FA15 Abstracts - SIAM · 2018. 5. 25. · 34 FA15 Abstracts NataliaC.Pinz´on-Cort´es...

32 FA15 Abstracts

IP1

Vector-Valued Nonsymmetric and Symmetric Jackand Macdonald Polynomials

For each partition τ of N there are irreducible representa-tions of the symmetric group SN and the associated Heckealgebra HN (q) on a real vector space Vτ whose basis isindexed by the set of reverse standard Young tableaux ofshape τ . The talk concerns orthogonal bases of Vτ -valuedpolynomials of x ∈ R

N . The bases consist of polyno-mials which are simultaneous eigenfunctions of commuta-tive algebras of differential-difference operators, which areparametrized by κ and (q, t) respectively. These polynomi-als reduce to the ordinary Jack and Macdonald polynomialswhen the partition has just one part (N). The polynomi-als are constructed by means of the Yang-Baxter graph.There is a natural bilinear form, which is positive-definitefor certain ranges of parameter values depending on τ , andthere are integral kernels related to the bilinear form forthe group case, of Gaussian and of torus type. The mate-rial on Yang-Baxter graphs and Macdonald polynomials isbased on joint work with J.-G. Luque.

Charles F. DunklUniversity of [email protected]

IP2

Two Variable q-Polynomials

Abstract not available at this time.

Mourad IsmailUniversity of Central [email protected]

IP3

On the Asymptotic Behavior of a Log Gas in theBulk Scaling Limit in the Presence of a VaryingExternal Potential

Abstract not available at time of publication.

Percy DeiftCourant InstituteNew York [email protected]

IP4

Title Not Available at Time of Publication

Abstract not available at this time.

Olga HoltzUniversity of California, BerkeleyTechnische Universitaet [email protected]

IP5

Integrable Probability and the Role of PainlevFunctions

We will review various models in probability that are inte-grable in the sense that various distribution functions canbe explicitly evaluated in terms of Painleve functions andtheir generalizations. We develop in more detail a classof stochastic growth models that belong to the Kardar-Parisis-Zhang (KPZ) universality class such as the asym-

metric simple exclusion process and the KPZ equation. Inaddition, the experiments of Takeuchi and Sano will bebriefly discussed.

Craig A. TracyUniversity of California, [email protected]

IP6

Limits of Orthogonal Polynomials and Contrac-tions of Lie Algebras

In this talk, I will discuss the connection between superin-tegrable systems and classical systems of orthogonal poly-nomials in particular in the expansion coefficients betweenseparable coordinate systems, related to representations ofthe (quadratic) symmetry algebras. This connection al-lows us to extend the Askey scheme of classical orthogonalpolynomials and the limiting processes within the scheme.In particular, for superintegrable systems in 3D, the poly-nomial representations of quadratic algebras are given interms of two-variable polynomials and the two-variableanalog of the Askey scheme, including the quadratic Racahalgebra, will be discussed along with the limiting processeswithin the scheme.

Sarah PostUniversity of Hawaii at ManoaDepartment of [email protected]

IP7

A New Look at Classical Orthogonal Polynomials

There are two possible definitions of classical orthogo-nal polynomials:(i) they satisfy a second order differentialor difference equation;(ii) (generalized) derivative of themgives again orthogonal polynomials. Both definitions arerelated with concrete forms of corresponding operators. Wepropose a new approach dealing with some abstract um-bral operators. This gives a wide generalization of a notionof classical orthogonal polynomials.

Alexei ZhedanovDonetsk Institute for Physics and Engineering, [email protected]

IP8

Asymptotic and Numerical Aspects of SpecialFunctions

For the numerical evaluation of special functions, asymp-totic expansions are an important tool. The standard ex-pansions can be used rather straightforwardly. The so-called uniform expansions need more attention, especiallyfor critical values of secondary parameters in the asymp-totic problem. For example, the Airy-type expansion of theBessel function Jν(z) can be used for large domains of theargument and order, but for the transition value z = ν spe-cial methods are needed for computing the coefficients. Wemention several methods for handling this type of problem.We start with a few examples for which Maple and Mathe-matica have problems in the evaluation of well-known spe-cial functions, like the Kummer U -function, for medium-sized values of the parameters. We discuss recent activitiesin the Santander-Amsterdam project on the evaluation ofspecial functions, in particular for certain cumulative dis-tribution functions. We start with the incomplete gamma

FA15 Abstracts 33

functions, and we give recent results for the non central chi-squared or the non central gamma distribution, also calledMarcum Q−function in radar detection and communica-tion problems. This is joint work with Amparo Gil andJavier Segura (University of Cantabria, Santander, Spain).

Nico M. TemmeCentrum Wiskunde & InformaticaThe [email protected]

IP9

Multivariate Orthogonal Polynomials and ModifiedMoment Functionals

Multivariate orthogonal polynomials can be defined bymeans a measure defined on a domain on R

d. A very impor-tant class of multivariate orthogonal polynomials is calledclassical because the measure satisfies a matrix analogueof the Pearson differential equation as well as the orthogo-nal polynomials are the eigenfunctions of a partial secondorder differential operator. In this talk, we present old andnew results on classical multivariate orthogonal polynomi-als. In particular, some classical multivariate orthogonalpolynomials and some useful modifications will be studied,as well as their impact into the useful properties of theorthogonal polynomials. We study the so–called Uvarovmodification obtained by adding to the measure one or afinite set of mass points. Recently, Christoffel modifica-tion in several variables, that is, the modification obtainedby multiplying the measure times a polynomial, has beenstudied in the frame of linear relations.

Teresa E. PerezUniversidad de GranadaDepartamento de Matematica [email protected]

IP10

Hypergeometric Series: On Number Theory’s Se-cret Service

A natural outcome of the theory of generalized hypergeo-metric functions are rational approximations to the valuesof Riemann’s zeta functions and alike mathematical con-stants. In my talk I plan to outline the way it goes (hyper-geometric series, hypergeometric Barnes- and Euler-typeintegrals) and stress on some recent achievements— thecurrent best irrationality measures of π (due to Salikhov),of log 2 (due to Marcovecchio) and of ζ(2). The final partof the talk will discuss some linear and algebraic indepen-dence results that make use of generalized hypergeometricfunctions.

Wadim ZudilinUniversity of [email protected]

IP11

Orthogonal Polynomials and the 2-Species ASEP

The asymmetric exclusion process (ASEP) is a model ofparticles hopping on a 1-dimensional lattice with openboundaries. The partition function of this model is relatedto moments of Askey-Wilson polynomials. Askey-Wilsonpolynomials are at the top of the hierarchy of orthogonalpolynomials, and are also a limiting case of Koornwinder

polynomials, which are multivariate orthogonal polynomi-als. Recently Eric Rains defined moments of Koornwinderpolynomials at q=t, which appear to be polynomials withpositive coefficients when written appropriately in the pa-rameters of the ASEP. I’ll explain joint work with SylvieCorteel in which we show that Koornwinder moments atq=t are related to the 2-species ASEP, an exclusion processinvolving two different types of particles. I’ll also describecomplementary work of Olga Mandelshtam and Xavier Vi-ennot providing a combinatorial description of the station-ary distribution of the 2-species ASEP.

Lauren WilliamsUniversity of California [email protected]

SP1

SIAG/OPSF Gbor Szeg Prize Announcement andLecture

Abstract not available at time of publication..

Karl LiechtyDePaul UniversityDepartment of Mathematical [email protected]

CP1

Killip-Simon Problem and Jacobi Flow on GsmpMatrices

D. Damanik, R. Killip and B. Simon completely describedthe spectral properties of Jacobi matrices J+, which are ina sence �2 perturbations of the isospectral torus of periodicJacobi matrices. The spectrum of a periodic Jacobi matrixis a system of intervals of a very specific nature. Jointlywith P. Yuditskii, we generalize this result to spectral sets,which are arbitrary finite system of intervals.

Benjamin EichingerInstitute of Analysis, Johannes Kepler University [email protected]

CP1

A Matrix Approach for the Semiclassical and Co-herent Orthogonal Polynomials

We obtain a matrix characterization of semiclassical or-thogonal polynomials in terms of the Jacobi matrix as-sociated with the multiplication operator in the basis oforthogonal polynomials, and the lower triangular matrixthat represents the orthogonal polynomials in terms of themonomial basis of polynomials. We also provide a matrixcharacterization for coherent pairs of linear functionals.

Lino G. GarzaUniversidad Carlos III de [email protected]

Francisco MarcellanUniversidad Carlos III de MadridInstituto de Ciencias Matematicas (ICMAT)[email protected]

Luis E. GarzaDepartamento Matematicas,Universidad [email protected]

34 FA15 Abstracts

Natalia C. Pinzon-CortesUniversidad Nacional de [email protected]

CP1

An Explicit Family of Askey-Wilson Type Matrix-Valued Orthogonal Polynomials

We discuss an explicit set of matrix-valued orthogonalpolynomials, which are eigenfunctions to a matrix-valuedAskey-Wilson type q-difference operator. A study of thecorresponding matrix-valued weight leads to an expres-sion of the matrix entries of these polynomials in termsof polynomials from the q-Askey-scheme. Other propertiesof these polynomials as well as an outlook will be discussed.(Joint work with Noud Aldenhoven and Pablo Roman.)

Erik Koelink, Noud AldenhovenRadboud [email protected], [email protected]

Pablo RomanUnivesidad Nacional de [email protected]

CP1

Jacobi Polynomial Moments and Products of Ran-dom Matrices

In this talk we study global singular value distributionsarising from products of complex Gaussian random matri-ces and truncations of Haar distributed unitary matrices.To this end, we introduce an appropriate general class ofmeasures with moments essentially given by specific Jacobipolynomials with varying parameters. Solving the under-lying moment problem is based on a study of the Riemannsurfaces associated to a class of algebraic equations.

Thorsten NeuschelDepartment of Mathematics, KU LeuvenLeuven, [email protected]

CP2

Special Functions As Non-Uniform Steady-StateSolutions of a Mathematical Model DescribingFluid-Solute Transport in Peritoneal Dialysis

Recently a mathematical model for fluid and solute trans-port in peritoneal dialysis has been proposed (R. Chernihaet al, Int. J. Appl. Math. Comp. Sci., 2014, V.24, P.837-851). The model is based on a nonlinear system of partialdifferential equations for fluid and solute concentrations.The model was simplified in order to obtain exact formulaefor steady-state solutions, hence, exact solutions involvingBessel and hypergeometric functions for the fluid and so-lute fluxes are constructed. Biomedical interpretation ofmathematical results is provided.

Roman ChernihaSchool of Mathematical Sci. University of [email protected]

Jacek WaniewskiIBIB of Polish Acad. Sci.Warsaw, Poland

[email protected]

CP2

Approximate Controllability of An Impulsive Neu-tral Differential Equation with Deviating Argu-ment and Bounded Delay

In this paper we prove the approximate controllability of animpulsive neutral differential equation with deviated argu-ment and control parameter included in the nonlinear term.We use Schuader fixed point theorem and fundamental as-sumptions on system operators to prove the result; therebyremoving the need to assume the invertibility of a control-lability operator which fails to exist in infinite dimensionalspace if the generated semigroup is compact. We also givean example to illustrate our result. In this paper we studythe approximate controllability of the following problem

d[x(t) + g(t, xt)]

dt= A[x(t) + g(t, xt))] +Bu(t) + f(t, x(a(x(t), t)), u(t)),

x0(θ) = φ(θ), θ ∈ [−r, 0]x(t+k )− x(t−k ) = Ik(x(tk)), k = 1, ..., m,

.

Sanjukta DasIndian Institute of Technology, Roorkee,sanjukta [email protected]

Dwijendra PandeyIndian Institute of Technology, [email protected]

CP2

On the Study of Solution of Lorenz System UsingGeneralized Mittag-Leffler Function

In this talk, generalized Mittag-Leffler function of Prab-hakar (1971) and Shukla et al. (2009) are applied to solveapproximate and analytical solutions of nonlinear frac-tional differential equation system such as Lorenz system.The obtained results were compared with the results ofHomotopy perturbation method and Variational iterationmethod.

Ranjan K. JanaS. V. NATIONAL INSTITUTE OF TECHNOLOGY(SVNIT)SURAT, GUJARAT, [email protected]

CP2

Hypergeometric Functions and Chebyshev Polyno-mials: Explicit Solutions of 2-D Free BoundaryProblems in Groundwater Hydrology

Steady, phreatic, Darcian flows in hillslope hydrology [1]are generalized to bedrocks with faults, karst or zonal frac-turing. By conformal mappings and solution of Hilbertsboundary value problems, characteristic functions (deriva-tives of complex potential/coordinate) are expressed ascombinations of hypergeometric functions with three over-lapping convergence domains. If a part of impermeablesubstratum is a control variable the kernel of the Cauchy-type integral is expanded into a series of Chebyshev poly-nomials. Accuracy of integrations and series truncations isstudied. 1. Kacimov A.R., Obnosov Yu.V., Abdalla, O.,

FA15 Abstracts 35

Castro-Orgaz O. (2014) Groundwater flow in hillslopes: an-alytical solutions by the theory of holomorphic functionsand hydraulic theory. Applied Mathematical Modelling,doi:10.1016/j.apm.2014.11.016.

Anvar KacimovSultan Qaboos [email protected]

Yurii ObnosovKazan Federal University, RussiaInstitute of Mathematics and [email protected]

CP3

Approximation by a Generalization of theJakimovski-Leviatan Operators

We introduce a generalization of the Jakimovski–Leviatan-Kantorovich Type operators constructed by A. Jakimovskiand D. Leviatan (1969) and the theorems on convergenceand the degree of convergence are established. Further-more, we study the convergence of these operators in aweighted space of functions on a positive semi-axis.

Didem Aydin AriKirikkale UniversityKirikkale/ [email protected]

Sevilay Kirci SerenbayBaskent UniversityAnkara / [email protected]

CP3

Integrals Involving Powers of K

Let K denote the complete elliptic integral of the first kind.We investigate integrals of the form

∫ 1

0

f(x)K(x)ndx,

where f is an algebraic function. We show that, for allnatural n, there exist classes of f such that this integralmay be evaluated in closed form; e.g.

∫ 1

0

K(x)3dx =3Γ(1/4)8

1280π2.

These results are proven using Eisenstein series and Fourierseries, and are connected to multiple sums arising fromchemical lattices.

James G. WanSingapore University of Technology and Designjames [email protected]

CP3

Reproducing Kernels of Spherical Monogenics

On the space of spherical harmonics there exists a repro-ducing kernel that is given by a Gegenbauer polynomial.By going over to complex variables, one obtains a reproduc-ing kernel expressed as a Jacobi polynomial. In this talk,the space of hermitian monogenics, which is the space ofpolynomial bihomogeneous null-solutions of two complex

conjugated Dirac operators, is considered. The reproduc-ing kernel for this space is obtained and expressed in termsof Jacobi polynomials.

Michael Wutzig, Hendrik De Bie, Frank SommenGhent [email protected], [email protected],[email protected]

CP3

New Index Transforms with the Product of BesselFunctions.

In this talk we discuss new index transforms with the prod-uct of Bessel functions as kernels. In particular, we willinvestigate a generalization of the Lebedev transform withthe square of the Macdonald function. Mapping propertiesare studied, inversion formulas are proved. As applications,initial value problems for higher order partial differentialand difference equations are solved.

Semyon YakubovichUniversity of PortoFaculty of Sciences, Dept. of [email protected]

CP4

Representations for the Parameter Derivatives ofOrthogonal Polynomials in Two Variables

The aim of this paper is to obtain the parameter derivativerepresentations in the form of

∂Pn,k(λ;x, y)

∂λ=

n−1∑m=0

m∑j=0

dn,j,mPm,j(λ;x, y)+k∑

j=0

en,j,kPn,j(λ;x, y)

for some families of orthogonal polynomials in two variableswith respect to their parameters by using the parameter

derivatives of the classical Jacobi polynomials P(α,β)n (x)

where λ is a parameter and 0 ≤ k ≤ n; n, k = 0, 1, 2, ....Furthermore, the orthogonality properties of the paramet-ric derivatives of these polynomials are discussed.

Rabia AktasAnkara [email protected]

CP4

Sums of Squared Baskakov Functions

The complete monotonicity of sums of squares of gener-alized Baskakov basis functions is proved by deriving thecorresponding result for certain hypergeometric functions.This implies its logarithmic convexity. Thereby a recentconjecture of Ioan Rasa is established. Moreover in thecentral Baskakov case the limit distribution of the zeros iscomputed for large values of a parameter. As a byproductnew representations for the complete elliptic integral of thefirst kind are established.

Wolfgang GawronskiDepartment of MathematicsUniversity of [email protected]

Ulrich AbelTechnical University of Applied Sciences MittelhessenGermany

36 FA15 Abstracts

[email protected]

Thorsten NeuschelDepartment of Mathematics, KU LeuvenLeuven, [email protected]

CP4

On the Horizontal Monotonocity of the GammaFunction

The behaviour of the gamma function along vertical linesin the right half plane has been studied closely but very lit-tle on the modulus of the gamma function along horizontallines in the upper half plane. We show that |Γ(t + ia)| ismonotone in t as soon as the imaginary part a exceeds athreshold value. We give a proof of this and indicate somesharp lower bounds for a beyond which this monotonic be-haviour holds.

Gopala Krishna SrinivasanIndian Institute of Technology [email protected]

CP5

Orthogonal Polynomial Interpretation of Δ-Todaand Volterra Lattices

The correspondence between dynamics of Δ-Toda and Δ-Volterra lattices for the coefficients of the Jacobi opera-tor and its resolvent function is established. A methodto solve inverse problem –integration of Δ–Toda and Δ-Volterra lattices – based on Pade approximants and con-tinued fractions for the resolvent function is proposed. Themain ingredient are orthogonal polynomials which satisfyan Appell condition, with respect to the forward differ-ence operator Δ. It is shown that the Δ-Volterra lattice isrelated to the Δ-Toda lattice by Miura or Backlund trans-formations.

Ana FoulquieUniversidade de [email protected]

Ivan AreaUniversidad de [email protected]

Amilcar BranquinhoUniversidade de [email protected]

Eduardo GodoyUniversidad de [email protected]

CP5

State of the Art Visualizations of Complex Func-tion Data

The NIST DLMF contains function visualizations thatwould have been difficult, if not impossible, to envisionwhen Abramowitz and Stegun was published. We haveimproved quality and accessibility using mesh generation,approximation theory, and the latest 3D web graphics tech-nology to capture key features such as zeros, poles, andbranch cuts. We demonstrate our redesign of the function

visualizations using WebGL, a JavaScript API for render-ing 3D graphics in a web browser without a plugin.

Bonita V. Saunders, Brian Antonishek, Qiming WangNational Institute of Standards and Technology (NIST)[email protected], [email protected],[email protected]

Bruce MillerApplied and Computational Mathematics DivisionNational Institute of Standards and [email protected]

CP5

Extension of Generalized Mittag-Leffler Densityand Processes

Shukla and Prajapati [J. Math. Anal. Appl. No336(2007),797-811] extended the Mittag-Leffler function

and defined as Eγ,qα,β(z) =

∑∞n=1

(γ)qnΓ(αn+β)

· zn

n!, where

α, β, γ ∈ C,�(α) > 0,�(β) > 0,�(γ) > 0, q ∈ (0, 1) ∪ N,and the function Eγ,q

α,β(z) converges absolutely for all z ∈ C

if q < �(α) + 1 (entire function of order �(α)−1) and for|z| < 1 if q = �(α)+1. In this paper, we obtain some prop-erties of Mittag-Leffler density, Structural representationof the Mittag-Leffler variable and Mittag-Leffler Stochas-tic Process by using Eγ,q

α,β(z).

Pratik V. ShahC.K.Pithawalla College of Engg. & Technology,Surat - 395007 , Gujarat , [email protected]

Ajay ShuklaS.V.National Institute of Technology,Surat 395007, Gujarat, [email protected]

CP5

Associated Polynomials, Markov’s Theorem, andFirst-Hitting Times of Birth-Death Processes

A classical result of Karlin and McGregor states that for abirth-death process on the nonnegative integers with corre-sponding birth-death polynomials {Qn}, the Laplace trans-form of the first-hitting time Tij satisfies

E[esTij ] =Qi(s)

Qj(s), s < 0,

provided i < j. In a recent paper (J. Theor. Probab. 25(2012) 950-980) Gong, Mao and Zhang, using the theoryof Dirichlet forms, established representations for E[esTij ]with i > j, featuring associated polynomials of {Qn}. Itwill be shown in the talk that these results may also beobtained with the help of Markov’s theorem.

Erik A. Van DoornUniversity of [email protected]

CP6

Ring of Integrals Operators

we define an ring of linear operators whose kernels are es-sentially Bessel functions. These operators act on spaces ofentire functions of exponential type and are endowed with

FA15 Abstracts 37

a complete and explicit symbolic calculus that we will de-scribed. We introduce a new symbols to obtain a new classpolynomials.

Miloud AssalInstitut Superieur des Mathematiques [email protected]

CP6

Stable Regions of Turan Expressions

Consider polynomial sequences that satisfy a first-order dif-ferential recurrence. We prove that if the recurrence isof a special form, then the Turan expressions for the se-quence are weakly Hurwitz stable (non-zero in the openright half-plane). A special case of our theorem settlesa problem proposed by S. Fisk that the Turan expressionsfor the univariate Bell polynomials are weakly Hurwitz sta-ble. We obtain related results for Chebyshev and Hermitepolynomials, and propose several extensions involving La-guerre polynomials, Bessel polynomials, and Jensen poly-nomials associated to a class of real entire functions. Thisis joint work with Lukasz Grabarek (University of Alaska)and Mirko Visontai (Google Reseach).

Matthew ChasseRochester Institute of [email protected]

CP6

New Characterizations of Leonard Pairs

Leonard pairs are pairs of linear transformations that acton each other’s eigenspaces in an irreducible tridiagonalfashion. They are related to the Askey scheme of orthogo-nal polynomials, distance-regular graphs, and the represen-tation theory of Lie algebras. In this talk, we will discusstwo new characterizations of Leonard pairs using certainsequences of parameters.

Edward HansonState University of New York at New [email protected]

CP6

A Classification of the Lowering-Raising Triples.

Let F denote a field, and let V denote a vector space overF with positive dimension d+1. By a decomposition of Vwe mean a sequence of one-dimensional subspaces whosedirect sum is V . Let {Vi}di=0 denote a decomposition ofV . A linear transformation A ∈ End(V ) is said to lower

{Vi}di=0 whenever AVi = Vi−1 for 1 ≤ i ≤ d and AV0 = 0.The map A is said to raise {Vi}di=0 whenever AVi = Vi+1

for 0 ≤ i ≤ d− 1 and AVd = 0. A pair of elements A,B inEnd(V ) is called lowering-raising (or LR) whenever thereexists a decomposition of V that is lowered by A and raisedby B. A triple of elements A,B,C in End(V ) is called LRwhenever any two of A,B,C form an LR pair. We classifyup to isomorphism the LR triples on V . We discuss howLR triples are related to the quantum group Uq(sl2).

Paul M. TerwilligerMath Department, University of Wisconsin-Madison

[email protected]

CP7

A Q-Extension of a Reduction Formula of Watson

In this paper, we give a reduction formula for the q-integral

In =

∫ 1

0

xα(qx; q)β Dnq−1

[xγ(qβ+1x; q)δ

]dqx, n = 0, 1, 2, ..

where α, β, γ, and δ are complex numbers. The reduc-tions formula is a three term recurrence relations of In.The representation of In in terms of basic hypergeometricseries will be investigated. This is a q-analogue of the workintroduced by W. N. Bailey in 1953.

Zeinab MansourKing Saud University Facuty of Science, MathematicsDep.Riyadh P.O. Box 2455 Riyadh 11451 Saudi [email protected]

Maryam AL TowailebKingdom of Saudi ArabiaP.O.Box 22459, Riyadh [email protected]

CP7

Some Q-Continued Fractions and Their Connec-tions with Lambert Series and Mock Theta Func-tions

Mock theta function is last gift to Mathematical worldby S.Ramanujan,that he quoted in his last letter to G.H.Hardy on January 12,1920. Lambert series is a well knownseries used by Ramanujan to prove many of his identities.Following Ramanujan, many mathematicians like GeorgeAndrews and B.C Berndt also used these series to provemany of Ramanujan identities. In the present work, bymaking use of Andrews and Hickerson identities, we haveestablished certain new q-continued fractions for the ratioof Lambert series and also for the ratio of combination ofmock theta function of order six.

Mohan RudravarapuGovernment Polytechnic,SrikakulamAndhra Pradesh,[email protected]

Pankaj SrivastavaMotilal Nehru National Instituite ofTechnology,[email protected]

CP8

On Verblunsky Coefficients Related to a ParticularClass of Caratheodary Functions

Schur algorithm provides a procedure to obtain Szego poly-nomials from Caratheodary functions defined on the unitdisk by means of PC-fractions. The Verblunsky coefficientsobtained from these Szego polynomials are useful in char-acterizing various interesting properties of these polyno-mials. In this work, a class of Verblunsky coefficients re-lated to a particular class of Caratheodary functions avail-able in the literature are considered. The correspondingChain sequences and minimal parameter sequences relatedto these Verblunsky coefficients are investigated. Using

38 FA15 Abstracts

a well-known procedure, the corresponding para orthogo-nal polynomials and Szego polynomials are reconstructedwhich lead to interesting observations. Various related ap-plications for these coefficients are obtained and comparedwith the results exist in the literature.

Swaminathan AnbhuI.I.T. [email protected], [email protected]

CP8

On Zeros of a Class of Real Self-Reciprocal Poly-nomials

We investigate the distribution, the simplicity and themonotonicity properties of the zeros around the unit circleand real line of the real self-reciprocal polynomials givenby Rn(z) = 1+d(z+z2+ · · ·+zn−1)+zn, where d is a realnumber and n > 0. The sequence of polynomials {Rn}satisfies a three term recurrence relation, this recurrencerelation is used to get further results.

Cleonice F. BraccialiUNESP - Universidade Estadual PaulistaDept of Applied [email protected]

Vanessa Botta, Junior PereiraUNESP - Universidade Estadual [email protected], [email protected]

CP8

Orthogonal Polynomials with As Their WeightFunctions

Abstract. The orthogonal polynomials are used very oftenin applied mathematics M. Podisuk and his colleague pre-sented some kinds of orthogonal polynomials in I, II, IIIand IV. In this paper, the orthogonal polynomials in theclosed interval [0,1] with the weight functions of the formwill be illustrated.

Maitree -. PodisukKasem Bundit [email protected]

CP8

Discrete Orthogonal and q-Orthogonal PolynomialSequences in the Extended Hahn Classes with Sim-ple Recurrence Coefficients

In [1] we found explicit formulas for the coefficients an andbn of the recurrence pn+1(x) = (x− bn)pn(x)− anpn−1(x)for all the elements in the extended Hahn classes of discreteorthogonal and q-orthogonal polynomial sequences. Suchcoefficients are rational functions of n, in the first case,and of qn, in the second case, which are determined by theinitial terms b0, b1, b2, a1, a2. We consider the cases thatyield simple rational functions. In addition to the well-known classical sequences we obtain many other examples,such as bn = b0 and an = a1n

2 or an = 3n4a1(4n2−1)−1, in

the discrete case. We also find the Bochner-type operatorfor each example. [1] L. Verde-Star, Linear Alg. Appl. 440(2014) 293–306.

Luis Verde-StarUniversidad Autonoma Metropolitana

[email protected]

CP9

Generalized Hurwitz Matrices: Criteria of TotalPositivity

Given an infinite-dimensional generalized Hurwitz matrixHM(f) = {aMj−i}ij , defined by a polynomial f(x) =a0x

n + a1xn−1 + . . . + an, ai ∈ R and a positive integer

M ≤ n, we provide a necessary and sufficient criterion forits total positivity. Our criterion is given in the form ofa finite number of determinantal conditions. We also con-sider a factorization of a totally positive generalized Hur-witz matrix using a generalization of the classical Euclideanalgorithm.

Olga Y. KushelTechnische Universitaet [email protected]

Olga HoltzUniversity of [email protected]

Sergey KhrushchevKazakh-British Technical Universitysvk [email protected]

Mikhail TyaglovShanghai Jiao Tong [email protected]

CP9

Canonical Vector-Polynomials for Complex OrderModified Bessel Functions

The canonical vector-polynomials for the approximation ofmodified Bessel functions of the second kind with com-plex order and real argument are constructed. LanczosTau method computational scheme is applied for the ap-proximation of the solutions of hypergeometric type dif-ferential equations and their systems. The stable recur-rent scheme is developed for the calculation of the canon-ical vector-polynomials coefficients. Some applications formixed boundary value problems in wedge domains areshown.

Juri M. RappoportRussian Academy of [email protected]

CP9

Products of Truncated Unitary Matrices

We study the joint singular value density of fixed or ran-dom matrices multiplied with truncated unitary matricesdistributed by the induced Haar measure. We prove thatthe squared singular values of a fixed matrix multipliedwith a truncated unitary matrix are distributed accordingto a polynomial ensemble, which is a special type of deter-minantal point process. As a corollary, we apply this resultto the situation where we multiply a truncated unitary ma-trix with a random matrix whose squared singular valuesare a polynomial ensemble and show that this structureis preserved by the product. As an application we derivethe joint singular value density of a product of truncatedunitary matrices and its kernel of the k-point correlationfunction. The correlation kernel can be written as a double

FA15 Abstracts 39

contour integral and using this integral representation weobtain hard edge scaling limits.

Dries StivignyKU [email protected]

CP9

Some Applications of Determninants to OrthogonalPolynomials

We show how different type of structered matrices and theirdetermiants contribute to the theory orthogonal polynomi-als and related ordinary differential equations and providenew classes of orthogonal polynomials.

Mikhail TyaglovShanghai Jiao Tong [email protected]

MS1

Multivariate Orthogonal Polynomials and Inte-grable Systems

In this talk we discuss multivariate orthogonal polynomialsin the Euclidean space and multivariate orthogonal Lau-rent polynomials in the unit torus. Using the Gauss-Borelfactorization of a moment matrix we consider three termrelations, Christoffel-Darboux formulas, Darboux transfor-mations and the connection with a hierarchy of compatiblenonlinear partial differential and difference equations of in-tegrable type.

Manuel ManasDepartamento de Fsica Teorica IIUniversidad Complutense de [email protected]

MS1

Orthogonal Polynomials on the Unit Ball andFourth Order Partial Differential Equations

In this work we study a family of mutually orthogonal poly-nomials on the unit ball with respect to an inner productwhich includes an additional spherical term. First, we willdeduce connection formulas relating classical multivariateorthogonal polynomials on the ball and the new family oforthogonal polynomials. Then, using the representation ofthese polynomials in terms of spherical harmonics, alge-braic, analytic and differential properties will be deduced.

Miguel PinarUniversity of [email protected]

MS1

Algebraic Interpretation of Multivariate q-Krawtchouk Polynomials

An algebraic interpretation of q-analogs of the multivari-ate Krawtchouk polynomials introduced by Tratnik will bepresented. It will be shown that these polynomials in nvariables arise as matrix elements of unitary ”q-rotation”operators expressed as q-exponentials in slq(n + 1) gener-ators realized in terms of the ladder operators of n inde-pendent q-oscillators. The main ideas will be illustrated by

focusing on the case n = 2.

Vincent GenestCentre de recherches mathematiquesUniversite de [email protected]

Sarah I. PostCentre de Recherches [email protected]

Luc VinetCentre de recherches [email protected]


MS2

Repeated Integrals of the Coerror Function, Revis-ited

Nonstandard Gaussian quadrature is applied to evaluatethe repeated integral inerfc x of the coerror function for n ∈N0, x ∈ R in an appropriate domain of the (n, x)-plane.Relevant software in Matlab is provided, in particular tworoutines evaluating the function to an accuracy of twelveresp. thirty decimal digits.

Walter GautshiPurdue [email protected]

MS2

Numerical Multivariate Polynomial Factorization

We present a method to extract the factors of a multivari-ate polynomial in floating point arithmetic. We establishthe connection between the irreducible factors of a multi-variate polynomial and the Taylor expansion of its recipro-cal. Based on this connection, a multivariate generalizationof the qd-algorithm is employed to compute the irreduciblefactors. Our iterative method does not require the inputpolynomial to be square-free. Moreover, this approach pro-vides additional insight to some problems in multilinearalgebra.

Wen-shin Lee, Annie CuytUniversity of [email protected], [email protected]

MS2

DLMF Standard Reference Tables on Demand(DLMF Tables) (6)

Most current libraries and systems produce tables of func-tion values with limited information about accuracy. Toaddress this void, NIST ACMD and the University ofAntwerp CMA Group are collaborating to build the DLMFStandard Reference Tables web service to provide a stan-dard of comparison for testing numerical software by com-puting, on demand, special functions to user-defined accu-racy with guaranteed error bounds. We will discuss thebeta version recently released, plus current and proposedwork.

Bonita V. Saunders

40 FA15 Abstracts

National Institute of Standards and Technology (NIST)[email protected]


Marjorie McClainMathematical and Computational Sciences DivisionNational Institute of Standards and [email protected]

Daniel LozierNational Institute of Standards and [email protected]

Andrew DienstfreyNational Institute of Standards and Technology (NIST)[email protected]

Annie CuytUniversity of [email protected]

Stefan Becuwe, Franky BackeljauwUniversity of Antwerp, [email protected],[email protected]

MS2

Near-Minimal Cubature Rules on the Disk

High-dimensional integration rules have received a lot ofattention in a whole series of paper. The connection withmultivariate orthogonal polynomials is usually present. Wepresent a unifying theory, including several of the existingminimal rules, departing from the rather recent concept ofspherical orthogonality. Use of the newly defined multivari-ate orthogonal functions leads to a structured set of Prony-structured equations from which the nodes and weights ofcertain minimal cubature formulae can be computed.

Irem YamanGebze Technical [email protected]

Annie CuytUniversity of [email protected]

Brahim BenouahmaneUniversite Hassan [email protected]

MS3

Bressoud Polynomials and the Rogers-RamanujnanIdentities

In 1983, D.M. Bressoud published An Easy Proof of theRogers-Ramanujan Identities (J. Number Th., 16(1983),235-241). The elegance of his proof has been noticed bymany. The conclusion of the paper is devoted to a general-ization of his result where he proves a polynomial identitywhich in the limit yields two instances of the generalizedRogers-Ramanjuan series/product identity (G.E. Andrews,

Proc. Nat. Acad. Sci., 71(1974), 4082-4085). A study ofthis final result in Bressoud’s paper reveals the Bressoudpolynomials which in turn have various applications andlead to open problems.

George E. AndrewsPenn State UniversityDepartment of [email protected]

MS3

Monodromy of Hypergeometric Functions in Sev-eral Variables

The analytic continuation properties of single variable hy-pergeometric values is well-known nowadays. The nextchallenge is to obtain similar results for the class ofA-hypergeometric functions. These were introduced byGel’fand, Kapranov and Zelevinsky in the late 1980’s andcontain the classical Appell and Lauricella functions as spe-cial cases.

Frits BeukersUtrecht [email protected]

MS3

q-Bessel Functions and Rogers-Ramanujan TypeIdentities

We show how q-Bessel functions of the first kind lead toRogers-Ramanujan type identities. Carlotta considered aspecial case of this and used it to give new proofs of theRogers-Ramanujan identities.

Mourad IsmailUniversity of Central [email protected]

Ruimimg ZhangNorthwest A & F [email protected]

MS3

Mahler Measures of Hyperelliptic Families

I will overview some recent progress on the Mahler mea-sure of two-variable polynomials corresponding to specialelliptic families; these are known as Boyd’s conjectures. Amachinery, which was created in our joint work with MatRogers and further developed by Anton Mellit and FrancoisBrunault, allows one to relate the Mahler measures to theL-values of the underlying elliptic curves when the latterare parametrised by modular units. The talk will culminatewith new results on Boyd’s conjectures for a hyperellipticfamily obtained in joint work with Marie Jose Bertin (Paris6).


MS4

Equilibrium Measures and Their Support

Let Q(x) be an admissible external field. Let ν be thesigned equilibrium associated with Q(x). Suppose that thepositive part of ν consists of finitely many intervals, and ν

FA15 Abstracts 41

has log-concave density there. We show that the support ofthe equilibrium measure consists of finitely many intervals.

David BenkoUniverity of South AlabamaMobile, [email protected]

MS4

Asymptotically d-Energy Minimizing Sequences ofConfigurations on d-dimensional Conductors

We show that the sequence of configurations on a compactset A ⊂ Rd with Ld(A) > 0 and Ld(∂A) = 0 (Ld is theLebesgue measure), obtained by intersecting A with anyd-dimensional lattice scaled by a factor going to zero isasymptotically d-energy minimizing. When Ld(∂A) > 0,an asymptotically d-energy minimizing sequence is con-structed. When A is contained in a d-dimensional C1 man-ifold, any asymptotically best-packing sequence of N-pointconfigurations is asymptotically d-energy minimizing.

Sergiy BorodachovTowson UniversityBaltimore, [email protected]

Douglas Hardin, Edward SaffVanderbilt [email protected], [email protected]

MS4

Covering and Separation for Points on the Sphere

The covering radius of an N-point set XN on the unitsphere in Rd+1 is the radius of the largest spherical cap thatcontains no points from XN whereas the separation dis-tance gives the least distance between two different pointsin XN . In the theory of approximation and interpola-tion, the covering radius arises in the error of approxima-tion whereas good separation of points is generally associ-ated with the stability of an approximation or interpola-tion method. We present resent results and open questionsregarding the covering and separation properties of, in par-ticular, random points on the sphere. This talk is basedon joint work with Josef Dick, Ed Saff, Ian Sloan, YuguangWang and Rob Womersley.

Johann BrauchartTechnical University at GrazGraz, [email protected]

MS4

Characterization of Complex Variable Positive Def-inite Functions

A holomorphic positive definite function f defined on a hor-izontal strip of the complex plane is characterized as theFourier-Laplace transform of a unique exponentially finitemeasure on R. This characterization unifies the integralrepresentations in Bochner’s theorem on positive definitefunctions and Widder’s theorem on exponentially convexfunctions. Special interesting cases include the Gammafunction and the Riemann Zeta function. Further charac-terizations are derived from the usual concepts of moment,

moment-generating function and characteristic function.

Jorge C. BuescuFaculdade de CienciasUniversidade de [email protected]

Antonio [email protected]

Alexandra [email protected]

MS5

On Perturbations of the Toda Lattice

We present the results of an analytical and numerical study(with Nenciu) of the long-time behavior for certain Fermi-Pasta-Ulam lattices viewed as perturbations of the com-pletely integrable Toda lattice. Our main tools are thedirect and inverse scattering transforms for doubly-infiniteJacobi matrices. We also present our work (with Trogdon)where we solve the Cauchy problem for the Toda lattice bycomputing the Riemann-Hilbert formulation of the inversescattering transform.

Deniz Bilman, Irina NenciuUniversity of Illinois at [email protected], [email protected]

Thomas TrogdonCourant Institute of Mathematical [email protected]

MS5

Transition Asymptotics for the Painleve II Tran-scendent

We consider real-valued solutions u = u(x|s), x ∈ R ofthe second Painleve equation uxx = xu + 2u3 whichare parametrized in terms of the monodromy data s ≡(s1, s2, s3) ⊂ C

3 of the associated Flaschka-Newell systemof rational ODEs. Our analysis describes the transitionasymptotics as x → −∞ between the oscillatory power-like decay asymptotics for |s1| < 1 (Ablowitz-Segur) to thepower-like growth behavior for |s1| = 1 (Hastings-McLeod)and from the latter to the singular oscillatory power-likegrowth for |s1| > 1 (Kapaev). It is shown that the transi-tion asymptotics is of Boutroux type, i.e. it is expressed interms of Jacobi elliptic functions.

Thomas J. BothnerUniversite de [email protected]

MS5

On Non-Linearizable Boundary Value Problems forthe Defocusing Nonlinear Schrodinger Equation onthe Half-Line

We study non-linearizable Dirichlet initial/boundary-valueproblems for the defocusing nonlinear Schrodinger equa-tion on the half-line x > 0 using a modification of theunified transform method. We sidestep the global relationby instead introducing an explicit approximation of the

42 FA15 Abstracts

Dirichlet-to-Neumann mapping for the problem suggestedby dispersionless theory. Accuracy is proven directly in thesemiclassical/dispersionless limit, with the help of steepestdescent methods. This is joint work with Zhenyun Qin,Fudan University.

Peter D. MillerUniversity of Michigan, Ann [email protected]

Zhenyun QinFudan [email protected]

MS5

Numerical Inverse Scattering for the BenaminOnoEquation

Numerical inverse scattering has proven a successfultool for calculating the Korteweg–de Vries and nonlinearSchrdinger equations, which consists of calculating spec-tral data for a differential equation (direct scattering)and solving a Riemann–Hilbert problem (inverse scatter-ing). The Benjamin–Ono equation does not fit immedi-ately in this form, as the direct scattering is a Riemann–Hilbert/differential equation and the inverse scattering isa nonlocal Riemann–Hilbert problem involving an oscilla-tory integral term. In this talk we investigate numerics forthese more difficult problems.

Sheehan OlverThe University of [email protected]

MS6

Multivariate Meixner Polynomials and a DiscreteModel of the Two-dimensional Harmonic Oscillatorwith su(2) Symmetry

A discrete model of the two-dimensional harmonic oscilla-tor based on the two-variable Meixner polynomials is pre-sented. It is shown that the system is superintegrable andhas su(2) symmetry. The interpretation of the d-variableMeixner polynomials as matrix elements for SO(d, 1) rep-resentations on oscillator states is reviewed.

Julien GaboriaudUniversite de [email protected]


Jessica LemieuxUniversity of [email protected]


MS6

Recent Developments in Hyperinterpolation

Hyperinterpolation is an approximation technique devel-oped by Ian Sloan in 1995. In this talk, a survey of recent

results in the field will be presented.

Jeremy WadePittsburg State [email protected]

MS6

A New Identity for Gegenbauer Polynomials andReproducing Kernels for Multivariable OrthogonalPolynomials

A new integral identity that expresses the Gegenbauerpolynomial Cλ

n in terms of an integral of Cμn , with μ > λ,

is established. It is used to derive concise formulas for re-producing kernels of orthogonal polynomials for two fam-ilies of weight functionsL: the product Gegenbauer weightfunctions on the unit cube and the generalized Gegenbauerweight functions on the unit ball of Rd.

Yuan XuUniversity of [email protected]

MS6

Some Topics on Basis Function Approximation onthe Sphere

In several cases, kernel-based methods for scattered datanicely frame within the context of approximation onGelfand pairs. With the Euclidean motion group beingthe most prominent example, another geometrical settingis given by zonal basis function approximation. The sphereis tied to the pair (SOd, SOd−1) with Gegenbauer polyno-mials defining the spherical functions. Aiming at transfer-ring well-established results from radial basis function ap-proximation to the geometric setting of the sphere, certainproperties of Gegenbauer polynomials are needed. Whilesome of these properties are already known, others lead tointeresting problems dealing with this family of orthogo-nal polynomials. We will address these issues and explainsome of the results. The talk is based on joint work withR. Beatson and Y. Xu.

Wolfgang zu CastellHelmholtz Zentrum Muenchen, German Research Centerfor EnvirIngolstaedter Landstrasse 1, 85764 Neuherberg, [email protected]

MS7

Algorithms for An Interpolation Problem on theUnit Circle Between Lagrange and Hermite Prob-lems

This piece of work is devoted to obtain formulae for aninterpolation problem on the Unit Circle (T). The interpo-lation conditions gather the values of the polynomial at 2nnodes and its first derivative at n nodal points; the nodalsystem is constituted by 2n equispaced points of modulus1. We give different solutions, between them we point outan explicit solution in terms of the natural basis and twobarycentric type formulae. All these formulae are suitablefor numerical applications.

Elias Berriochoa, Alicia CachafeiroUniversity of [email protected], [email protected]

Jose GarcıA-Amor

FA15 Abstracts 43

I. E. S. Valle Inclan, Xunta de [email protected]

MS7

Bernoulli Polynomials As a Base in NumericalMethods

In theory, since the best approximation in L2[a, b] is unique,all polynomials which are in L2[a, b] can give the same ap-proximation of known functions. However, in application,it does not occur because we have computational error. Weknow usually Legendre polynomials have the less compu-tational error than other polynomials. We show Bernoullipolynomials create less computational error than Legendrepolynomials. For this reason, we introduced the Bernoullipolynomials as trial functions.

Somayeh MashayekhiMississippi state [email protected]

MS7

Approximation of Periodic Functions in Terms ofModuli of Continuity

A 2p periodic function in Lp spaces is approximated bythe trigonometric polynomials of degree and the error ofapproximation is given by norm minimozation of the norm||Tn(x) − f(x)||p. . The error defined above depends onthe various properties of the function f. In this paper, weobtain the error of approximation in terms of the modulusof continuity of the function f. We use some summabilitytechniques to accelerate the rate of convergence. We alsoderive some corollaries from our results.

Uaday SinghIndian Institute of Technology RoorkeeRoorkee-247667 (India)[email protected]

Soshal SainiIndian Institute of Technology [email protected]

MS7

A New Way to Compute Sine Integral Function

Considering its wide application areas in science and engi-neering, efficient calculation of the sine integral function isessential. We present a new method to calculate the sineintegral function which is based on an accurate represen-tation of the sinc function as a sum of scaled cosines. Theproposed method is as accurate as MATLAB’s sinint andis more efficient than both MATLAB’s sinint and trun-cated spherical Bessel function expansion, especially forlarge number of points.

Evren [email protected]

Garret FlaggSchlumberger, Houston, TX

[email protected]

MS8

Boundary Integrals and Approximations of Har-monic Functions

Steklov expansions for a harmonic function on a rectangleare analyzed. The value of a harmonic function at the cen-ter of a rectangle is shown to be well approximated by themean value of the function on the boundary plus a verysmall number (often 3 or fewer) of additional boundaryintegrals. Similar approximations are found for the cen-tral values of solutions of Robin and Neumann boundaryvalue problems. These results are based on finding explicitexpressions for the Steklov eigenvalues and eignfunctions.This is joint work with Professor Giles Auchmuty.

Giles AuchmutyUniversity of Houston, [email protected]

Manki ChoUniversity of [email protected]

MS8

Universal Lower Bounds for Potential Energy ofSpherical Codes

Minimal energy configurations (codes), maximal codes,and spherical designs have wide ranging applications invarious fields of science, such as crystallography, nanotech-nology, material science, information theory, wireless com-munications, etc. We derive universal lower bounds for thepotential energy of spherical codes, that are optimal in theframework of the standard linear programming approachand provide a characterization of when improvements arepossible. Our bounds are universal in the sense of Cohnand Kumar; i.e., they apply to any absolutely monotonepotential.

Peter DragnevIndiana UniversityPurdue University Fort [email protected]

Peter BoyvalenkovInstitute for Mathematics and Informatics, Sofia, [email protected]

Douglas Hardin, Edward SaffVanderbilt [email protected],[email protected]

Maya StoyanovaSofia University, [email protected]

MS8

Asymptotics of Minimal Discrete Periodic EnergyProblems

Let L be a d-dimensional lattice in Rd. For a parameter s >0, we consider the asymptotics of N point configurationsminimizing the L-periodic Riesz s-energy as the number ofpointsN goes to infinity. In particular, we focus on the case0 < s < d of long-range potentials where we establish thatthe minimal energy Es(L,N) is of the form Es(L,N) =

44 FA15 Abstracts

C0N2 +C1N

1+s/d + o(N1+s/d) as N → ∞. The constantsC0 and C1 depend only on s, d, and covolume of the latticeL.

Douglas HardinVanderbilt [email protected]

MS8

Asymptotics for Maximal Polarization Configura-tions

For a compact manifold A in Euclidean space and a lowersemi-continuous kernel K on A× A we consider the prob-lem of determining the asymptotic behavior of N-pointconfigurations {xi}Ni=1 that maximize the minimum of the

potential∑N

i=1K(x, xi) for x in A. We refer to such ex-tremal problems as polarization problems (the terminologyChebyshev problems is also used). We focus especially onRiesz s-kernels K(x, y) = 1/|x − y|s for s > 0 and theirconnection with best covering problems on A. Results fors > dim(A) are of particular interest.

Edward SaffVanderbilt [email protected]

MS9

Saturated Bands in Many-Pole Riemann-HilbertProblems

Many-pole Riemann-Hilbert problems arise in the analysisof semiclassical limits for differential equations and orthog-onal polynomials. In current methods certain parameterranges corresponding to saturated bands must be excludedfor technical reasons. We present progress on obtainingasymptotic results in these regions with applications to thenonlinear Schrodinger equation, the sine-Gordon equation,and discrete orthogonal polynomials.

Robert J. BuckinghamDept. of Mathematical SciencesThe University of [email protected]


David A. SmithUniversity of [email protected]

Megan StoneUniversity of [email protected]

MS9

Long Time Asymptotics and Stability of FiniteDensity Solutions of the Defocusing NLS Equation

We consider the Cauchy problem for the defocusing nonlin-ear Schrodinger equation (NLS). Using the ∂ generalizationof the nonlinear steepest descent method of Deift and Zhouwe derive the leading order approximation to the solutionof NLS in the solitonic region of space–time |x| < 2t forlarge times and provide bounds for the error which decaysas t ↗ ∞ for a general class of initial data whose differ-

ence from the non vanishing background possess’s a fixednumber of finite moments and derivatives. Using proper-ties of the scattering map of NLS we derive as a corollaryan asymptotic stability result for initial data which are suf-ficiently close to the N-dark soliton solutions of NLS.

Robert JenkinsDept. of MathematicsUniversity of [email protected]

MS9

Long-Time Asymptotics for a Discrete-Discrete In-tegrable Equation

The lattice potential KdV equation is a fully discrete inte-grable equation, whose initial value problem was recentlystudied by Butler and Joshi. We obtain long-time asymp-totics for this equation, using a suitable modification ofthe methods of Teschl’s group. The results show a strik-ing similarity with corresponding results for the continuousKdV equation, reinforcing the value of this discretization.

David A. SmithUniversity of [email protected]

MS9

Direct Scattering and Small Dispersion for theBenjamin-Ono Equation with Rational Initial Data

We propose a construction procedure for the scatteringdata of the Benjamin-Ono equation with a rational initialcondition, under mild restrictions. The construction pro-cedure entails building the Jost solutions of the Lax pairexplicitly and use their analyticity properties to recoverthe reflection coefficient, eigenvalues, and phase constants.We finish by showing that this procedure validates certainwell-known formal results obtained in the zero-dispersionlimit.

Alfredo N. Wetzel, Peter D. MillerUniversity of Michigan, Ann [email protected], [email protected]

MS10

An Historical Approach to Sobolev OrthogonalPolynomials

We present a survey about analytic propeties of polyno-mials orthogonal with respect to a Sobolev inner prod-uct. The distribution of their zeros, asymptotic proper-ties, spectral theory as well as some applications will beconsidered. Some open problems will be discussed.

Francisco MarcellanUniversity of Carlos III of [email protected]

MS10

Linearly Related Sequences of Continuous and Dis-crete Derivatives of Orthogonal Polynomial Se-quences. Connections with Sobolev Orthogonal

FA15 Abstracts 45

Polynomial Sequences

In this talk we consider structure relations of the form

N∑i=0

ri,nDmQn−i+m(x) =

M∑i=0

si,nDkPn−i+k(x) ,

where (Pn)n and (Qn)n are two sequences of orthogonalpolynomials (OPs), continuous or discrete, and, for a non-negative integer number s, Ds represents the continuousderivative or a discrete difference-derivative operator of or-der s (the possibilities for D include the usual discrete op-erators Dω and Dq). In the above relation, the orders ofthe derivatives, k and m, are arbitrarily fixed nonnegativeinteger numbers, and (ri,n)n and (si,n)n are sequences ofnumbers fulfilling some appropriate conditions. Such struc-ture relation leads to the notion of (M,N)−coherence oforder (m, k), being an extension of the concept of coher-ence of measures as introduced by A. Iserles, P. E. Koch,S. P. Nørsett, and J. M. Sanz-Serna [On polynomials or-thogonal with respect to certain Sobolev inner products,J. Aprox. Theory, 65(2) (1991) 151-175]. We will presenta survey of some results obtained in the last years aroundthe concept of (M,N)−coherence of order (m, k). In thecontinuous case, we will be able to present some exam-ples showing how such a concept may be an useful tool inApproximation Theory to explore polynomials orthogonalwith respect to a Sobolev inner product of the form

〈f, g〉λ,s :=

∫ +∞

−∞fg dμ+ λ

∫ +∞

−∞f (s)g(s) dν ,

provided (Pn)n and (Qn)n are orthogonal with respect topositive Borel measures μ and ν (resp.). Most resultspresented in this talk are the result of several collabora-tions along the last years, specially with F. Marcellan, R.Alvarez-Nodarse, M.N. de Jesus, N.C. Pinzon-Cortes, andR. Sevinik-Adıguzel.

Jose Carlos S. PetronilhoDepartamento de Matematica, Universidade de [email protected]

MS10

(M,N)-Coherent Pairs of Linear Functionals andJacobi Matrices

We give a matrix interpretation of (M,N)-coherent pairs oflinear functionals: An algebraic relation between the corre-sponding Jacobi matrices associated with such functionalsis stated, and the classical case is analyzed; The relation be-tween the Jacobi matrices associated with (M,N)-coherentpairs of orderm and the Hessenberg matrix associated withthe multiplication operator in terms of the basis of polyno-mials orthogonal with respect to the Sobolev inner productdefined by such coherent pair is deduced.

Natalia C. Pinzon-CortesUniversidad Nacional de [email protected]


MS10

W (1,p) - Convergence of Fourier-Sobolev ExpansionsAssociated to (M,N) Coherent Pairs of Measures

Let {qn,λ}n≥0 be the sequence of polynomials orthonormal

withrespect to the Sobolev inner product

〈f, g〉S :=

∫ 1

−1

f(x)g(x)dμ0(x) + λ

∫ 1

−1

f ′(x)g′(x)dμ1(x),

where λ is a non-negative constant and (dμ0, dμ1) is an(M,N)-coherent pair of measures supported on the interval[−1, 1] with M and N fixed non-negative integer numbers.In this talk we discuss necessary and/or sufficient condi-tions for the convergence in the W 1,p([−1, 1], (dμ0, dμ1))norm of the Fourier expansion in terms of {qn,λ}n≥0, with1 < p <∞.

Alejandro J. UrielesDepartamento de Matematicas, Universidad del [email protected]

MS11

Fast Arbitrary-precision Evaluation of SpecialFunctions in the Arb Library

We demonstrate Arb, a library for arbitrary-precision realand complex ball/interval arithmetic with support formany special functions, including erf, gamma, zeta, poly-log, Bessel, hypergeometric, theta, elliptic and modularfunctions. Our implementations are competitive with ex-isting non-interval high-precision software, and often sig-nificantly faster. One highlight is a new algorithm for el-ementary functions, giving up to an order of magnitudespeedup at ”medium” precision (approximately 200-2000bits).

Fredrik JohanssonInstitut de Mathematiques de BordeauxUniversite de Bordeaux 351, cours de la Liberation - [email protected]

MS11

Pushing Forward the Dimension of fcc Lattices

The generating function Gd for the return probabilities in ad-dimensional face-centered cubic lattice satisfies an ODEwhose order grows quadratically with d. Until recently onlythe ODEs for d ≤ 7 were known; using a recursive methodfor computing the coefficients of Gd, proposed by Zenine,Hassani, Maillard, we are able to go up to d = 11. TheseODEs share many remarkable properties which we shalldiscuss in this talk. This is joint work with Jean-MarieMaillard.

Christoph KoutschanRICAM LinzAustrian Academy of Sciences (AW)[email protected]

MS11

Divisibility Properties of Sporadic Apery-likeNumbers

It was shown by Gessel that the Apery numbers, intro-duced by Apery in his proof of the irrationality of ζ(3), areperiodic modulo 8. We investigate this, and other divisibil-ity results, for Apery-like numbers. For instance, we provethat the Almkvist–Zudilin numbers are periodic modulo 8as well. The ingredients of our proof are a multivariate ra-tional function whose diagonals are the Almkvist–Zudilinnumbers and a theorem originating with Furstenberg which

46 FA15 Abstracts

states that, modulo a fixed prime power, these values havealgebraic generating function and, hence, can be generatedby a finite state automaton. As demonstrated recently byRowland and Yassawi, these automata can be computedmechanically. This talk includes joint work with ArianDaneshvar, Pujan Dave, Amita Malik and Zhefan Wangthat was done as part of an Illinois Geometry Lab project.

Armin StraubUniversity of Illinois at [email protected]

MS11

Positive Rational Functions and their Diagonals

An interesting phenomenon about some arithmetically sig-nificant sequences, like the sequence of Apery numbers(http://oeis.org/A005259), is that they are the diagonalsequences of the Taylor expansions of “interesting’ rationalfunctions, and the positivity of the coefficients in the latterexpansions are (conjecturally) determined by the positivityof the sequences. In my talk I will explain and illustratethe phenomenon and also relate it to other classical resultsin analysis, in particular, to some 60-year old theorems ofHormander. This is joint work with Armin Straub.


MS12

Mock Modular Forms of Weight 5/2 and Partitions

We study the coefficients of a natural basis for the spaceof mock modular forms of weight 5/2 on the full modulargroup. The “shadow’ of the first element of this infinitebasis encodes the values of the partition function p(n). Weshow that the coefficients of these forms are given by tracesof singular invariants. These are values of modular func-tions at CM points or their real quadratic analogues: cycleintegrals of such functions along geodesics on the modu-lar curve. The real quadratic case relates to recent workof Duke, Imamoglu, and Toth on cycle integrals of the j-function, while the imaginary quadratic case recovers thealgebraic formula of Bruinier and Ono for the partitionfunction.

Nickolas AndersenUniversity of Illinois at [email protected]

MS12

Partition Identities and Mock Theta Functions

This is joint work with Stephen Hill, a Penn State under-graduate. In 1961, Basil Gordon proved a sweeping gener-alization of the Rogers-Ramanujan identities. His theoremmay be broadly characterized as identifying the generat-ing function for partitions having specified difference con-ditions on the parts with the quotient of two theta func-tions. We shall provide a new class of partitions (similarto those studied by Gordon) where the generating functionis identified with the quotient of a Hecke-type theta seriesdivided by the Dedekind eta function. The simplest caseis related to one of the fifth order mock theta functions ofRamanujan. The partitions in question are similar in kindto those described in: G.E Andrews, Partitions with initialrepetitions, Acta Math. Sinica, English Series, 25(2009),

1437-1442, and in G.E Andrews, Partitions with early con-ditions, In Advances in Combinatorics Waterloo Workshopin Computer Algebra, W80 May. 26-29, 2011.

George E. AndrewsPenn State UniversityDepartment of [email protected]

MS12

Mock Theta Functions, Partial Theta Functions,and Their Ghosts - Rhoades

There is an association between mock theta functions andpartial theta functions; they each carry half of the Fouriercoefficients of a non-holomorphic modular form. In thisassociation some mysterious q-hypergeometric series mayarise. We call these ”ghost terms”. This talk explains theassociation and gives hints of the curious structure in ghostterms which remains without a theory.

Robert RhoadesStanford [email protected]

MS12

Mock Theta Functions and Quantum ModularForms

In this talk, I will describe several related recent results re-lated to mock theta functions. These functions have veryrecently been understood in a modern framework thanksto the work of Zwegers and Bruinier-Funke. Here, we willrevisit the original writings of Ramanujan and look at hisoriginal conception of these functions, which gives rise toa surprising picture connecting important objects such asgenerating functions in combinatorics and quantum mod-ular forms.

Larry RolenUniversity of [email protected]

MS13

Painleve Equations - Nonlinear Special Functions

The Painleve equations, discovered over a hundred yearsago, are special amongst nonlinear ordinary differentialequations in that they are “integrable” due to their rep-resentation as Riemann-Hilbert problems. The Painleveequations can be thought nonlinear analogues of the clas-sical special functions and have numerous remarkable prop-erties. In this talk I shall give an introduction to thePainleve equations and discuss some open problems.

Peter ClarksonUniversity of KentKent [email protected]

MS13

Numerics for Classical Applications of Riemann-Hilbert Problems

We overview several classical problems that can be re-duced to RiemannHilbert problems, falling into three cat-egories: integral representations, differential equations andinverse spectral problems. The integral representation of

FA15 Abstracts 47

error functions and elliptic integrals can be rewritten interms of scalar RiemannHilbert problems. The Stokes’phenomenon for Airy’s equation and monodromy prob-lems for Fuchsian differential equations lead to matrix Rie-mannHilbert problems. The inverse spectral problem forJacobi and Schrdinger operators can be solved via matrixRiemannHilbert problems depending on a parameter, thelatter of which encodes the solution to the KortewegdeVries equation. In all three cases, applying numerics tothe RiemannHilbert problem allows for efficient approxi-mation, that is uniformly accurate in the complex plane.Joint work with Thomas Trogdon

Sheehan OlverThe University of [email protected]

MS13

Explorations of the Solution Space of the FourthPainlev Equation

Solutions to the Painleve equations (PI -PV I) that are freeof poles over extensive regions of the complex plane areof considerable interest, dating back to the tronquee andtritronquee solutions of PI (Boutroux 1913). This talkpresents parameter choices and initial conditions of PIV

that exhibit these pole-free sectors, highlighting the exis-tence of the families of solutions that are free from (or haveonly a finite number of) poles on the real axis.

Jonah A. [email protected]

Bengt FornbergUniversity of [email protected]

MS13

Uniformly Accurate Computation of Painleve IITranscendents

The Riemann–Hilbert approach for the Painleve transcen-dents has proved to be a powerful and rigorous tool forasymptotic analysis. The approach makes use of the Deift–Zhou method of nonlinear steepest descent. Recently, thisapproach has been adapted for numerical purposes and theresulting computations are seen to be uniformly accurate.In this talk, I will outline the method and describe the rig-orous justification for the accuracy that is observed. Thisis joint work with Sheehan Olver.


MS14

The Statistical Behaviour of the Low-end Spectraof Several Coupled Matrices and the Meijer-G ran-dom Point Field

Universality results in the statistical behaviour of spectraof random matrices typically consider the fluctuations neara macroscopic point of the spectrum. In an ensemble ofpositive definite matrices, there are two essentially differ-ent distinguished points; the largest eigenvalues and the

smallest. Since they are positive definite the study of thesmallest eigenvalue is the study of the statistics near theorigin of the spectral axis. The simplest instance is the La-guerre ensemble; in this case the fluctuations are describedby a determinant random point field defined in terms ofthe celebrated Bessel kernel. This behaviour is also provenin the literature to be ?universal?. We consider a model ofseveral coupled matrices of Laguerre type (with a specificinteraction) and address the corresponding study of theorigin of the spectrum. We find a natural generalization ofthe Bessel random point field to a multi-specie analog thatinvolves special functions (Meijer-G). It is then natural toformulate a universality conjecture.

Marco BertolaConcordia University, [email protected]

Thomas BothnerCRM, Concordia [email protected]

MS14

The Normal Matrix Model in the SupercriticalRegime, and Asymptotics of the Associated Or-thogonal Polynomials

The normal matrix model is a random matrix model witheigenvalues in the complex plane. The average character-istic polynomials are orthogonal with respect to weightedarea measure on the full plane. However, the integrals thatare involved in the orthogonality do not converge in manyinteresting cases, and then the orthogonality has to be re-defined. We follow the approach of [Bleher and Kuijlaars,Adv. Math. (2012)] for the model with a cubic potential,where the orthogonality is redefined as orthogonality oncontours in the complex plane. The orthogonal polynomi-als admit a Riemann-Hilbert characterization of size 3x3,which was analyzed in the subcritical case by means of theDeift-Zhou method of steepest descent. Here we analyzethe supercritical case, and we find the limiting behavior ofzeros. We also find a new critical behavior, and our resultsonly hold up to this second criticality.

Arno KuijlaarsKatholieke Universitaet [email protected]

Alexander TovbisUniversity of Central FloridaDepartment of [email protected]

MS14

Discrete Toeplitz Determinants and Their Appli-cations

We will discuss the asymptotics of the Toeplitz determi-nants with discrete measure. We first show how one canconvert this problem to the asymptotics of continuous or-thogonal polynomials by using a simple identity. Then weapply this method to the width of non-intersecting pro-cesses of several different types.

Zhipeng LiuCourant Institute of Mathematical SciencesNYU

48 FA15 Abstracts

[email protected]

MS14

The Condition Number of the Critically–Scaled La-guerre Unitary Ensemble

In recent computations with S. Olver, we demonstrateduniversal fluctuations in the iteration count, called the halt-ing time, of classical numerical algorithms with randominitial data. In particular, we noticed this phenomenon inthe conjugate gradient algorithm for solving Ax = b whereA > 0. The random data is formed by letting A = XX∗

where X is an N × n matrix with iid entries distributedaccording to some random variable D and n is scaled in acritical manner. In this talk, I will present a limit theoremfor the condition number of the Laguerre Unitary Ensem-ble (D is a complex Gaussian) with this critical scalingand relate it to the performance of the conjugate gradientalgorithm. This is joint work with P. Deift and G. Menon.


MS15

Differential Equations for Discrete Sobolev Orthog-onal Polynomials

The aim of this talk is to study differential properties oforthogonal polynomials with respect to a discrete Sobolevbilinear form with mass points. We will focus on the La-guerre case, where the mass point is located at zero, andthe Jacobi case, where the mass point can be located eitherat -1 or 1, or at the two points at the same time. We con-struct the orthogonal polynomials using certain Casoratideterminants. Using this construction, we prove that theyare eigenfunctions of a differential operator. Moreover, theorder of this differential operator is explicitly computed interms of the matrices which defines the discrete Sobolevbilinear form.

Manuel Dominguez de la IglesiaInstituto de Matematicas, Universidad NacionalAutonoma [email protected]

Antonio DuranUniversidad de [email protected]

MS15

Laguerre Polynomials and Sobolev Orthogonality

When α > −1, the classical Laguerre polynomials{Lα

m}∞m=0 form a complete orthogonal set in the Hilbertspace L2((0,∞);xαe−x). More generally, from classicalproperties of these polynomials, it is the case that, for eachnon-negative integer n, these Laguerre polynomials form acomplete orthogonal set in a certain Sobolev spaceWn withinner product

(f, g)n =n∑

j=0

∫ ∞

0

S(j)n f (j)(x)g(j)(x)xα+je−xdx (f, g ∈ Wn),

where {S(j)n } are the Stirling numbers of the second kind.

When α ≤ −1 but −α is not a positive integer, the La-guerre polynomials are orthogonal with respect to a signed

measure.A natural question to ask is what happens in the remainingcase when α is a real number: namely when −α := k is apositive integer? In this case, it is not difficult to see thatthe Laguerre polynomials {L−k

m | m ≥ k} form a complete

orthogonal set in the Hilbert space L2((0,∞);x−ke−x). Anot-so-obvious question to ask is whether there is a ‘natu-ral’ inner product in which the entire sequence of Laguerrepolynomials {L−k

m }∞m=0 are orthogonal? The answer, sur-prisingly, is yes. We will show that the Laguerre polynomi-als {L−k

m }∞m=0 form a complete orthogonal set in a Sobolevspace Sk with an inner product of the form

(f, g)Sk =k∑

j=0

∫ ∞

0

f (j)(x)g(j)(x)dμj.

We will also discuss a self-adjoint operator Tk, generated bythe classical second-order Laguerre differential expression,in the space Sk having the Laguerre polynomials {L−k

m }∞m=0

as a complete set of eigenfunctions.

Lance L. LittlejohnDepartment of Mathematics, Baylor Universitylance [email protected]

MS15

A Study of the Exceptional Xm-Jacobi Expressionfor Extreme Parameter Choices

In 2009, Gomez-Ullate, Kamran, and Milson extended thewell-established Bochner Classification (1929) by showingthat the only polynomial sequences {pn}∞n=1 which simul-taneously form a complete set of eigenstates for a second-order differential operator and are orthogonal with re-spect to a positive Borel measure having positive momentsare the exceptional X1-Laguerre and X1-Jacobi polynomi-als. We will focus on the exceptional X1-Jacobi polyno-mials. The second-order differential expression has ratio-nal coefficients and as a result, there is no solution of de-

gree zero. The X1-Jacobi polynomials{P

(α,β)n

}∞

n=1form

a complete orthogonal set in the weighted Hilbert spaceL2((−1, 1); wα,β), where wα,β is a positive rational weightfunction. Restrictions are placed on parameters α and β; inparticular, α, β > 0. We are particularly interested in the‘extreme’ parameter choice when α = 0. In this situation,{P

(α,β)n

}∞

n=2are in the associated L2 space, but we may

actually study the full sequence of solutions{P

(0,β)n

}∞

n=0in a certain Sobolev space S.

Jessica StewartDepartment of Mathematics and Computer Science,Goucher [email protected]

MS15

Orthogonal Polynomials for Learning

Within the burgeoning field of machine learning kernelbased learning has produced not only state of the art al-gorithms but stands out for the depth of its mathematicalsophistication and elegance. The promise of kernel learningfor applications lies in the abundance of reproducing ker-nels. In practice however there is a dearth of computablekernels. Orthogonal polynomials provide a compelling wayfor producing useful learning kernels. Associated spectraltheory gives key insight into the nature of the regulariza-tion properties for learning machines with Sobolev orthog-

FA15 Abstracts 49

onal polynomial kernels. Well discuss general theory aswell as several specific examples used in commercial appli-cations.

Richard WellmanDepartment of Mathematics, Westminster [email protected]

MS16

Transformations of Hypergeometric Functions

It is well known that the generalized hypergeometric func-tions pFq satisfy many useful functional identities includ-ing the Kummer, Pfaff or Euler transform. Less is knownin higher dimensions. Some results of a similar kind areknown for example for Appell’s functions and other func-tions. We are going to provide a generalization of Euler,Kummer and “Analytic continuation’ transform for n di-mensional hypergeometric function for arbitrary n.

Petr BlaschkeMathematical Institut in Opava, Silesian university inOpavaCzech [email protected]

MS16

Computing the Probability of Collision for Short-term Space Encounters: a Symbolic-Numeric Ap-proach

The increasing number of space debris in Low Earth Or-bits constitute a serious hazard for operational satellites.In order to provide adequate collision avoidance strategies,it is important to determine the collision probability be-tween two orbiting objects. Three-dimensional Gaussianprobability densities represent the position uncertainties ofthe objects. With some simplifying assumptions, the prob-lem of computing the collision probability, for short-termencounters between space-borne objects, is, in practice, re-duced to a two-dimensional integral of a Gaussian functionover a bounded region in a plane normal to the relativevelocity vector (encounter frame). The method presentedhere is based on an analytical expression for the integral,derived by use of Laplace transform and properties of D-finite functions. The formula has the form of a productbetween an exponential term and a convergent power se-ries with positive P-recursive coefficients. Analytic boundson the truncation error are also derived. This allows foran efficient and reliable numerical evaluation of the risk.This talk is based on R. Serra, D. Arzelier, M. Joldes, J.-B. Lasserre, A. Rondepierre and B. Salvy, A New Method toCompute the Probability of Collision for Short-term SpaceEncounters, AIAA/AAS astrodynamics specialist confer-ence, American Institute of Aeronautics and Astronautics,pages 1-7, 2014.

Mioara JoldesLAAS-CNRS7 Avenue du Colonel Roche, 31077 Toulouse, Cedex [email protected]

MS16

The Positive Part of Multivariate Infinite Series

The positive part of an infinite series is defined as the for-mal power series which is obtained from it by discarding allthe terms involving negative exponents. In the univariate

case, it is easy to see that the positive part of a rationalseries is again a rational series. While this is no longer truein the multivariate case, it is still true there that the pos-itive part of a D-finite series is D-finite. This can be usedin combinatorics to show that certain generating functionsare D-finite. In the talk, we will show how to compute thepositive part of a D-finite multivariate Laurent series usingcreative telescoping. This is joint work with Alin Bostan,Frederic Chyzak, Lucien Pech and Mark van Hoeij.

Manuel KauersJohannes Kepler [email protected]

MS16

About Some Identities by Bayad and Beck Involv-ing Bernoulli-Barnes Numbers, Barnes Zeta Func-tions and Fourier Dedekind Sums

A. Bayad and M. Beck have recently derived new identitiesinvolving the Bernoulli-Barnes numbers, the Barnes zetafunctions and the Fourier Dedekind sums. Using symboliccalculus methods, we will provide some extensions and in-terpretations of these identities. We will also show a linkbetween the Bernoulli-Barnes numbers and the p-Bernoullinumbers as recently introduced by Rahmani, which are re-lated to the generating function of some special zeta values.

Christophe VignatDpt. of Physics, Universite d’OrsayDpt. of Mathematics, Tulane [email protected]

MS17

A Generalized Lebesgue Identity in Ramanujan’sLost Notebook

The Lebesgue identity is one of the important identitiesin the theory of partitions and q-hypergeometric series (q-series). The identity has a free parameter. By dilationsof the base q, and choices of the free parameter, severalfundamental partition identities follow. In Ramanujan’sLost Notebook there is an identity in two free parameters.By viewing this as an extension of the Lebesgue identity,we obtain new weighted partition theorems. Our approachyields new combinatorial information about certain Hecke-Rogers type series, and also leads to new companions toEuler’s celebrated Pentagonal Numbers Theorem.

Krishnaswami AlladiUniversity of [email protected]

MS17

Exotic Bailey-Slater Spt-Functions and Hecke-Rogers Double Series

We study SPT-crank type functions that arise from Baileypairs and that have interesting arithmetic properties. Wefind representations of these functions in terms of infiniteproducts or two-variable Hecke-Rogers double series. Themethod uses Bailey’s Lemma and conjugate Bailey pairs.

Frank Garvan, Chris Jennings-ShafferUniversity of Florida

50 FA15 Abstracts

[email protected], [email protected]

MS17

Rogers-Ramanujan Type Identities

The Rogers–Ramanujan identities and a number of closelyrelated identities first appeared just over 120 years ago.Early contributors to the theory of Rogers–Ramanujan-type identities included Rogers, Ramanujan, F. H. Jack-son, W. N. Bailey, F. J. Dyson, and L. J. Slater. Sincetheir introduction, identities of Rogers–Ramanujan typehave found applications in a variety of areas including thetheory of partitions, statistical mechanics, and vertex oper-ator algebras. I will review some of the history of Rogers–Ramanujan type identities, including some of the more re-cent contributions which relied rather heavily on computeralgebra.

Andrew V. SillsGeorgia Southern [email protected]

MS17

Partitions Associated with the Ramanujan/WatsonMock Theta Functions omega(q) and nu(q)

Recently, George Andrews, Atul Dixit, and I have discov-ered very interesting partition theorems that are related tothe mock theta functions ω(q) and ν(q). For instance, thegenerating function for partitions where each part is lessthan twice the smallest part equals q ω(q). In this talk, Iwill present those discoveries and some related arithmeticproperties.

Ae Ja YeeDepartment of MathematicsPennsylvania State [email protected]

George E. AndrewsPenn State [email protected]

Atul DixitTulane [email protected]

MS18

Zeros of Large Degree Vorob’ev-Yablonski Polyno-mials Via a Hankel Determinant Identity

It is well known that all rational solutions of the secondPainleve equation and its associated hierarchy can be con-structed with the help of Vorob’ev-Yablonski polynomi-als and generalizations thereof. The zero distribution ofthe aforementioned polynomials has been analyzed numer-ically by Clarkson and Mansfield and the authors observeda highly regular and symmetric pattern: for the Vorob’evpolynomials itself the roots form approximately equilat-eral triangles whereas they take the shape of higher or-der polygons for the generalizations. Very recently Buck-ingham and Miller completely analyzed the zero distribu-tion of large degree Vorob’ev-Yablonski polynomials usinga Riemann-Hilbert/nonlinear steepest descent approach tothe Jimbo-Miwa Lax representation of PII equation. In ourwork, joint with Marco Bertola, we rephrase the same prob-lem in the context of orthogonal polynomials on a contourin the complex plane. The polynomials are then analyzed

asymptotically and the zeros localized through the vanish-ing of a theta divisor on an appropriate hyperelliptic curve.Our approach starts from a new Hankel determinant rep-resentation for the square of the Vorob’ev-Yablonski poly-nomial. This identity is derived using the representationof Vorob’ev polynomials as Schur functions.

Thomas BothnerCRM, Concordia [email protected]

MS18

Asymptotics of Large-Degree Rational Painleve-IVFunctions

The Painleve-IV equation admits a family of rational so-lutions, indexed by two integers, that can be expressed interms of generalized Hermite polynomials. In the large-degree limit the zeros of these polynomials form remark-able patterns in the complex plane resembling rectangleswith arbitrary aspect ratios depending on how the indexingintegers grow. Using Riemann-Hilbert analysis we analyt-ically determine the boundary of the zero/pole region forthese rational functions.

Robert BuckinghamUniversity of [email protected]

MS18

Painleve I in the Cubic Random Matrix Model

In his talk we analyze the double scaling regime in theasymptotics of the partition function in the cubic randommatrix model. In particular, we show the appearance ofsolutions of the Painlev I differential equation in this set-ting. We also provide a uniform asymptotic expansionof the Painlev I function Ψ(ζ, λ, α) introduced by Kapaevto describe the Riemann-Hilbert problem corresponding toPainlev I.

Alfredo DeanoDepartamento de MatematicasUniversidad Carlos III de [email protected]

Pavel BleherDepartment of Mathematics, [email protected]

MS18

On Large Degree Rational Solutions of Painleve II

The inhomogeneous Painleve-II equation with parameterm has a (unique) rational solution exactly when m is aninteger. Motivated in part by the universal appearanceof these functions in a certain double-scaling limit involv-ing the semiclassical sine-Gordon equation, we study theirasymptotic behavior in the limit of large m by means ofsteepest descent methods applied to a Riemann-Hilbertproblem encoding them that naturally arises in the sine-Gordon theory. Joint work with Robert Buckingham(Cincinnati).


FA15 Abstracts 51

Robert BuckinghamUniversity of [email protected]

MS19

The Riemann-Hilbert Approach to PolynomialsOrthogonal with Respect to Complex Weight Func-tions

In this contribution we illustrate how the Riemann-Hilbertapproach and the Deift-Zhou steepest descent method canbe used to analyze the asymptotic behavior and zero dis-tribution of polynomials that are orthogonal with respectto complex weight functions. Some examples studied re-cently include a Fourier-type weight function, w(x) = eiωx

on [−1, 1], an exponential weight with a cubic potential onsuitable contours in the complex plane, and Bessel func-tions on the semiaxis [0,∞).

Alfredo DeanoDepartamento de MatematicasUniversidad Carlos III de [email protected]

MS19

The Riemann-Hilbert Approach to Critical Phe-nomena in the Two Matrix Model

In this talk, an overview will be presented on various resultsfor the asymptotic analysis of the two matrix model, basedon the Riemann-Hilbert approach for the biorthogonalpolynomials that integrate this model. Particular emphasiswill be on recent results on the classification of the criticalphenomena that can occur in the even quadratic/quarticcase and the even quartic/quartic case.

Maurice DuitsMathematics DepartmentStockholm [email protected]

MS19

Asymptotics of Orthogonal Polynomials with Com-plex Varying Weight: Critical Point Behaviour andthe Painleve Equations

We study the asymptotics of recurrence coefficients forcomplex monic orthogonal polynomials πn(z) with thequartic exponential weight

exp

[−n

(z2

2+tz4

4

)],

where t ∈ C and N → ∞. We consider neighborhoods ofthe critical points t0 = − 1

12, t1 = 1

15and t2 = 1

4, where the

subleading terms can be expressed via Painleve transcen-dents. These subleading terms can become dominant nearthe poles of the corresponding Painleve transcendents. Weuse the nonlinear steepest descent analysis for Riemann-Hilbert Problems to describe the recurrence coefficients infull neighborhoods of tj , j = 0, 1, 2, including the locationof the poles. We also provide the global (in the t-plane)“phase diagrams, where the recurrence coefficients exhibitdifferent asymptotic behaviors (Stokes’ phenomenon).

Alex TobvisDepartment of MathematicsUniversity of Central [email protected]

Marco BertolaConcordia University, [email protected]

MS19

Asymptotics in the Complex Plane for Multiple La-guerre Polynomials

There are two kinds of multiple Laguerre polynomials. Themultiple Laguerre polynomials of the first kind have or-thogonality conditions with respect to weights xαje−x on[0,∞) for 1 ≤ j ≤ r, where αj > −1 are such thatαi − αj /∈ Z. Multiple Laguerre polynomials of the secondkind have orthogonality properties with respect to xαe−cjx

on [0,∞) for 1 ≤ j ≤ r, where cj > 0 are such that ci �= cjwhenever i �= j. Wielonsky and Lysov (Constr. Approx.28 (2008), 61–111) have obtained strong asymptotics forthe multiple Laguerre polynomials of the second kind usingthe Riemann-Hilbert problem and the Deift-Zhou steep-est descent method. We will obtain strong asymptoticsof the multiple Laguerre polynomials of the first kind us-ing the same techniques. This is joint work with ThorstenNeuschel.

Walter van AsscheKatholieke Universitaet [email protected]

MS20

A Particular Case of a Higher Order Sobolev-TypeInner Product of Orthogonal Polynomials in Sev-eral Variables

We consider sequences of polynomials of several variables,orthogonal with respect to the Sobolev-type inner productgiven in, which is obtained by adding to an inner stan-dard product, the gradient operator of order j evaluatedin a particular point. We present an expression for theperturbed orthogonal polynomials in terms of the originalpolynomials and a particular example on the unit ball inwhich we analize the asymptotic behaviour of the Kernelassociated to the Sobolev-type polynomials.

Herbert DuenasDepartamento Matematicas, Universidad [email protected]

Wilmer M. GomezUniversidad Pedagogica y Tecnologica de [email protected]

MS20

Sobolev Orthogonal Polynomials on the Unit Ballvia Outward Normal Derivatives

The purpose of this work is to analyze a family of mutu-ally orthogonal polynomials on the unit ball with respectto an inner product which involves the outward normalderivatives on the sphere. Using the representation of thesepolynomials in terms of spherical harmonics, algebraic andanalytic properties will be deduced. First, we will get con-nection formulas relating classical multivariate orthogonalpolynomials on the ball with our family of Sobolev orthog-onal polynomials. Then explicit expressions for the normswill be obtained, among other properties.

Lidia Fernandez

52 FA15 Abstracts

Departamento Matematica Aplicada, [email protected]

Antonia DelgadoUniversidad de [email protected]

Teresa E. PerezUniversidad de GranadaDepartamento de Matematica [email protected]

Miguel PinarUniversity of [email protected]

MS20

On Asymptotic Properties of Sobolev OrthogonalPolynomials on the Unit Circle

In this contribution, we study the sequence of polynomialsorthogonal with respect to the Sobolev inner product

〈f, g〉S :=

∫T

f(z)g(z)dμ(z) + λf (j)(α)g(j)(α),

where μ is a nontrivial probability measure supported onthe unit circle, α is a complex number, λ is a positive realnumber, and j is a positive integer. In particular, we an-alyze some asymptotic properties of such polynomials andthe behavior of their zeros when n and λ tend to infinity,respectively. We also provide some numerical examples toillustrate the behavior of these zeros with respect to α.

Luis E. GarzaDepartamento Matematicas,Universidad [email protected]


Kenier CastilloUniversity Carlos III de [email protected]

MS20

Orthogonal Polynomials with Respect to a SobolevInner Product on the Unit Circle

We look at some information concerning the orthogonalpolynomials with respect to the Sobolev inner product

〈f, g〉S = (1− t) 〈f, g〉b + t f(1) g(1) + κ〈f ′, g′〉b+1,

where 0 ≤ t < 1, κ ≥ 0, Re(b) > −1/2 and

〈f, g〉b = τ (b)

2π

∫ 2π

0

f(eiθ) g(eiθ) (eπ−θ)Im(b)(sin2(θ/2))Re(b)dθ.

Here, τ (b) = 2b+b |Γ(b+1)|2Γ(b+b+1)

is such that 〈1, 1〉b = 1 and

hence also 〈1, 1〉S = 1. For example, the monic orthogonalpolynomials Sn with respect to the inner product 〈f, g〉Ssatisfy

Sn(z) + anSn−1(z) = Φn(z), n ≥ 1,

where Φn(z) are the monic orthogonal polynomials withrespect to the inner product 〈f, g〉b,t = (1 − t) 〈f, g〉b +

t f(1) g(1).

Alagacone S. RangaDepartamento de Matematica AplicadaComputacao,[email protected]

MS21

A Human Proof of Gessel’s Lattice Path Conjec-ture

Gessel walks are lattice paths confined to the quarter planethat start at the origin and consist of unit steps going ei-ther West, East, South-West or North-East. In 2001, IraGessel conjectured a nice closed-form expression for thenumber of Gessel walks ending at the origin. In 2008,Kauers, Koutschan and Zeilberger gave a computer-aidedproof of this conjecture. The same year, Bostan and Kauersshowed, again using computer algebra tools, that the com-plete generating function of Gessel walks is algebraic. Inthis talk, I will report on the first human proofs of thesetwo results, obtained in collaboration with Irina Kurkovaand Kilian Raschel. The proofs are derived from a newexpression for the generating function of Gessel walks interms of Weierstrass zeta-functions.

Alin BostanINRIA Saclay [email protected]

MS21

Numerical Evaluation of Contour Integrals forComputation of Stirling Numbers

Flajolet and Prodinger (SIAM J. Discrete Math, 12:155–159, 1999) proposed an interpolation of Stirling partitionnumbers by the integral definition{

xy

}=

(x− 1)!

(y − 1)!

1

2iπ

∫H

exp(z)(exp(z)− 1)y−1 dz

zx.

Because the Γ functions are singular at negative integersthis is not entirely satisfactory and some authors have pro-posed instead a related integral. This talk explores theseissues and discusses an explicit evaluation by numericalmeans, using a parametrization of the Hankel contour thatis related to the Tree T function (a cognate of LambertW more suited to combinatorial applications, as is wellknown).

Rob CorlessUniversity of Western Ontario in [email protected]

David JeffreyDepartment of Applied MathematicsUniversity of Western [email protected]

Yang WangUniversity of Western [email protected]

MS21

Utility Maximization and Symbolic Computation

We give a brief introduction to utility maximization, which

FA15 Abstracts 53

is a central topic of mathematical finance. Recently, therehas been a lot of work on extending the classical frame-work by modelling the bid-ask spread. For small spread,asymptotic expansions of the economically relevant quan-tities can be obtained. The special functions in our talkare the value functions of the optimization problems. Theyare characterized by Riccati ODEs with free boundary, andhave explicit expressions in some cases. Computer algebra,in particular symbolic manipulation of polynomials, is veryuseful to obtain asymptotic expansions of optimal strate-gies, optimal utility, and trading volume.

Stefan GerholdFinancial and Actuarial MathematicsVienna University of [email protected]

MS21

On 2F1-type Solutions of Second Order Linear Dif-ferential Equations

Differential equations with 2F1-type solutions are verycommon in Mathematics and they also occur in severalareas such as Physics and Combinatorics. Given a secondorder linear differential operator L ∈ C(x)[∂], we want tofind a 2F1-type solution of the form

exp(

∫r dx) · 2F1(a, b; c; f)

where r, f ∈ Q(x), and a, b, c ∈ Q. This form is bothmore and less general than in prior work. In prior work,solutions involving a sum of two 2F1’s were also considered,however, f was restricted to rational functions, while ourmethod allows algebraic functions.

Erdal ImamogluDepartment of MathematicsFlorida State [email protected]

Mark van HoeijFlorida State UniversityTallahassee, [email protected]

MS22

Ramanujan, Voronoi Summation Formula, Circleand Divisor Problems and Some Modular Trans-formations

On page 336 in his Lost Notebook, Srinivasa Ramanujanproposed an identity that may have been devised to attacka divisor problem. Unfortunately, the identity is vitiatedby a divergent series appearing in it. We present here a cor-rected version of Ramanujan’s identity. This study has anatural connection with the Voronoi summation formula.We also obtain a one-variable generalization of two dou-ble Bessel series identities of Ramanujan on page 335 ofthe Lost Notebook which are intimately connected withthe circle and divisor problems. Finally, we also obtain anew modular-type transformation involving infinite seriesof Lommel functions. Such a transformation is extremelyrare, and is the only known example of its kind.

Atul DixitTulane [email protected]

Bruce Berndt, Arindam Roy, Alexandru Zaharescu

University of Illinois at [email protected], [email protected], [email protected]

MS22

On Theta Quotients Generating Graded Algebrasof Modular Forms

We construct theta quotients that generate graded alge-bras of modular forms on principal congruence subgroupsof prime level. A variety of consequences ensue, includ-ing coupled systems of differential equations for sums oftwisted Eisenstein series analogous to those on the fullmodular group.

Tim HuberDepartment of MathematicsUniversity of Texas-Pan [email protected]

MS22

Cubic Modular Equations in Two Variables

By adding certain equianharmonic elliptic sigma func-tions to the coefficients of the Borwein cubic theta func-tions, an interesting set of six two-variable theta func-tions may be derived. These theta functions invert theF1

(13; 13; 13; 1|x, y) case of Appell’s hypergeometric function

and satisfy several identities akin to those satisfied by theBorwein cubic theta functions. Previous work on thesefunctions is extended and put into the context of Ramanu-jan’s modular equations, resulting in a simpler derivationsas well as several new modular equations for Picard mod-ular forms.

Dan SchultzPennsylvania State [email protected]

MS22

Special Values of Trigonometric Dirichlet Series

In his first letter to Hardy and in several entries of his note-books, Ramanujan recorded many evaluations of trigono-metric Dirichlet series such as

∞∑n=1

coth(πn)

n2r−1=

1

2(2π)2r−1

r∑m=0

(−1)m+1 B2m

(2m)!

B2(r−m)

(2(r −m))!,

which, in fact, goes back to Cauchy and Lerch. A recent ex-

ample is the secant Dirichlet series ψs(τ ) =∑∞

n=1sec(πnτ)

ns ,for which Lalın, Rodrigue and Rogers conjecture, and par-tially prove, that its values ψ2m(

√r), with r > 0 rational,

are rational multiples of π2m. We give an overview of spe-cial values of such Dirichlet series and their connection withthe theory of modular forms. This talk includes joint workwith Bruce C. Berndt.

Armin StraubUniversity of Illinois at [email protected]

MS23

Painleve Equations and Orthogonal Polynomials

In this talk I shall discuss semi-classical orthogonal poly-nomials arising from perturbations of classical weights. Itis shown that the coefficients of the three-term recurrence

54 FA15 Abstracts

relation satisfied by the polynomials can be expressed interms of Wronskians which involve special functions. TheseWronskians are related to special function solutions of thePainleve equations. Using this relationship recurrence rela-tion coefficients can be explicitly written in terms of exactsolutions of Painleve equations.


MS23

Determinantal Representations of Exceptional Or-thogonal Polynomials

Exceptional orthogonal polynomials represent an escape tothe Bochner Classification Theorem by not allowing poly-nomials of certain degrees as eigenfunctions to the corre-sponding differential expressions. Much of their fascinat-ing rich structure had been exposed over the past 6 (orso) years, and still more interesting research is being con-ducted. We focus on the exceptional X1−Laguerre poly-nomials, which do not contain the constant polynomial. Inparticular, we find determinantal representations in termsof moments where the first row of the determinant is ad-justed to implement the ’exceptional condition’. A slightalteration of the moments greatly simplifies the represen-tation.

Constanze LiawBaylor UniversityUSAconstanze [email protected]

MS23

On the Alternative Discrete Painleve I

During this talk I will discuss special solutions of the al-ternative discrete Painleve-I equation. At the centre willbe the uniqueness of the positive solution, which some-how links to the recurrence coefficients of exponential cubicweights.

Ana LoureiroUniversity of [email protected]


Walter van AsscheKatholieke Universitaet [email protected]

MS24

Orthogonal Polynomials for a Class of Measureswith Discrete Rotational Symmetries in the Com-plex Plane

Normal matrix models for external potentials of the form

|z|2n + tzd + tzd are considered with integers 0 ≤ d ≤ 2n.A symmetry reduction procedure is used to find the equi-librium measure for all values of n, d and t. For fixed nand d, there is a critical value |t| = tcr such that the sup-port is simply connected for |t| < tcr and has d connected

components for |t| > tcr. Moreover, the strong asymp-totics of orthogonal polynomials with respect to measures

of the form e−|z|2d+tzd+tzddA(z) are obtained, where t iscomplex and dA is the area measure on the plane. Sincethe orthogonality can be written in terms of contour in-tegrals, the asymptotics are found via a Riemann–Hilbertproblem. In particular, the Cauchy transform of the nor-malized counting measure of the zeroes converges to thatof the equilibrium measure in the exterior of the support(based on joint works with T. Grava and D. Merzi).

Ferenc BaloghSISSA [email protected]

MS24

Asymptotics for the Partition Function in Two-cutRandom Matrix Models

I will present a new method to obtain asymptotics for thepartition function in two-cut random matrix models, basedon Riemann-Hilbert problems. The method enables usto re-derive potential-dependent terms which were knownin the physics literature, but also to evaluate rigorouslypotential-independent terms. The talk will be based onjoint work with T. Grava and K. McLaughlin.

Tom ClaeysDepartment of [email protected]

Tom ClaeysUniversite catholique de [email protected]

MS24

Recent Developments in the Large-N Analysis ofCorrelation Functions in the Quantum Separationof Variables Method

The scalar products and certain correlation functions ofmodels solvable by the quantum separation of variables canbe expressed in terms of N-fold multiple integrals whichcan be thought of as the partition function of a one dimen-sional gas of particles evolving on a curve C, trapped inan external potential V and interacting through repulsive

two-body interactions of the type ln[sinh[πω1(λ − μ)] ·

sinh[πω2(λ − μ)]]. The choice of the curve C and of the

confining potential V determines a given model. The anal-ysis of the large-N asymptotic behaviour of these integralsis of interest to the description of the continuum limit ofthe integrable model. In this talk, I shall report on recentdevelopments in the large-N analysis of such integrals anddiscuss, on some specific examples, the form taken by theasymptotics. Part of the results that I will present issuefrom a joint work with G. Borot and A. Guionnet.

Karol K. KozlowskiInstitut de Mathematiques de [email protected]

MS25

Deformed Semi-classical Discrete Orthogonal Poly-nomials

In this talk a study families of discrete orthogonal poly-nomials on the real line whose Stieltjes functions satisfy a

FA15 Abstracts 55

linear type difference equation with polynomial coefficientsis presented. The discrete dynamical systems, obtained asa result of deformations of the recurrence relation coeffi-cients of the orthogonal polynomials related to the abovereferred Stieltjes functions is derived.

Amılcar BranquinhoUNIVERSIDADE DE [email protected]

MS25

Integral Transforms of d-orthogonal Polynomial Se-quences

This talk is mainly focused on index type integral trans-forms that map certain d-orthogonal polynomial sequencesinto another d-orthogonal polynomial sequences. It turnsout that in several cases the weights are semiclassical.

Ana LoureiroUniversity of [email protected]

MS25

The Correspondence between the Askey Tableof Orthogonal Polynomial Systems and the SakaiScheme of Discrete Painleve Equations

The Askey Table is a classification of the hypergeomet-ric and basic hypergeometric orthogonal polynomial sys-tems in a single variable, whose members possess a num-ber of characteristic and defining properties. On the otherhand the Sakai Scheme is a classification of the non-linear, integrable Painleve equations and their difference,q-difference and elliptic analogs, based upon algebraic-geometric ideas. It is possible to bring these two into cor-respondence whereby the Askey Table represents a base,trivial level and the Sakai Scheme is its first deformation,the first of the multi-variable extensions. In the process ofdoing so we solve a number of related problems: an algo-rithmic construction of Lax pairs for the discrete Painleveequations, the construction of an explicit sequence of clas-sical solutions to the discrete Painleve equations with use-ful applications to probabilistic models such as found inrandom matrix theory or random tiling models. In addi-tion the correspondence provides insight into a geometricalreformulation of the Askey Table itself akin to the root sys-tem symmetries of the hypergeometric functions.

Nicholas WitteUniversity of [email protected]

MS25

”Abstract” Classical Polynomials: Open Problems

We discuss open problems arising in classification of ”ab-stract” classical orthogonal polynomials. These polynomi-als satisfy some general ”umbral” eigenvalue problems withunknown symbols of operators.


MS26

Interlacing and Bounds for Zeros of Quasi-

Orthogonal Laguerre Polynomials

We discuss interlacing properties of zeros of polynomials

of consecutive degree in sequences of Laguerre polynomials

{L(α)n }∞n=0 characterized by −2 < α < −1. Stieltjes inter-

lacing between the zeros of qm and pn+1, m ≤ n, where{qn}∞n=0 is a sequence of quasi-othogonal Laguerre polyno-mials and {pn}∞n=0 is an orthogonal Laguerre sequence, isalso investigated. Upper and lower bounds for the negative

zero of L(α)n , −2 < α < −1, are derived.

56 FA15 Abstracts

Kathy A. DriverUniversity of Cape TownSouth [email protected]

Martin E. MuldoonDepartment of Mathematics & StatisticsYork [email protected]

MS26

Weighted Norm Inequalities for Some SpecialFunctions

A method of obtaining new weighted norm inequalities forgeneralized hypergeometric functions and special functionsof hypergeometric type will be presented. It is based onthe limit version of the sequence of weighted convolutioninequalities generated by a seminorm problem for formalpower series and its binomial solution. It will be shownthat the limit inequality involves some deep properties ofthe Eulerian integrals and Bernstein polynomial relation.Several related examples and applications of our methodwill be discussed.

Arcadii GrinshpanUniversity of South FloridaTampa, FL, [email protected]

MS26

Meijer’s G Function and Fox’s H Function Near aRegular Singularity

In the talk, we will discuss some new properties of Mei-jer’s G function Gp,0

p,p and Fox’s H-function Hp,0q,p . The first

function has been studied by various authors under dif-ferent names and our first goal is to reveal the connectionsbetween those investigations. In particular, we will presentnew formulas for the expansion coefficients in the vicinityof the singular point at 1. The Fox H-function does notsatisfy any known differential equation but it has a similartype of singularity at a finite real point (which need not beunity). We will discuss its behavior in the neighborhood ofthis point. Further, we present new integral and functionalequation for both functions and inequalities for some othercases of Meijer’s G-function. The talk will reflect the jointwork with Elena Prilepkina.

Dmitry KarpRussian Academy of SciencesVladivostok, [email protected]

MS26

Interlacing of Zeros of General Laguerre Polynomi-als

We consider interlacing properties of the real zeros of the

Laguerre polynomial L(α)n (x), in the case α < −1. The

main tool used is the Sturm comparison theorem.

Martin E. MuldoonDepartment of Mathematics & StatisticsYork [email protected]

Kathy A. DriverUniversity of Cape TownSouth [email protected]

MS27

The Quantum Superalgebra ospq(1|2) and a q-generalization of the Bannai-Ito Polynomials

The Racah problem for the quantum superalgebraospq(1|2) is considered. A quantum deformation of theBannai-Ito algebra is realized by the intermediate Casimiroperators entering the Racah problem. A q-generalizationof the Bannai-Ito polynomials is presented. The relationbetween these basic polynomials and the q-Racah/Askey-Wilson polynomials is discussed.




MS27

Separation of Variables, Superintegrability andBocher Contractions

Two-dimensional quadratic algebras are generalizations ofLie algebras that include the symmetry algebras of 2ndorder superintegrable systems in 2 dimensions as specialcases. Distinct superintegrable systems and their quadraticalgebras are related by geometric contractions, inducedby generalized Inonu-Wigner Lie algebra contractions andthese contractions have important physical and geometricconsequences, such as the Askey scheme for hypergeomet-ric orthogonal polynomials. This approach can be unifiedby ideas first introduced in the 1894 thesis of Bocher tostudy separable solutions of the wave equation.

Willard Miller JrUniversity of MinnesotaSchool of [email protected]

MS27

Exceptional Orthogonal Polynomials, Wronskians,and the Darboux Transformation

Exceptional orthogonal polynomials (so named becausethey span a non-standard polynomial flag) are definedas polynomial eigenfunctions of Sturm-Liouville problems.By allowing for the possibility that the resulting sequenceof polynomial degrees admits a number of gaps, we ex-tend the classical families of Hermite, Laguerre and Jacobi.In recent years the role of the Darboux (or the factoriza-tion) transformation has been recognized as essential in thetheory of orthogonal polynomials spanning a non-standardflag. In this talk we will focus on exceptional Hermite poly-nomials: their regularity properties, asymptotics of zerosand their relation to the recent conjecture that ALL excep-tional orthogonal polynomials are related via factorization

FA15 Abstracts 57

transformations to classical orthogonal polynomials.

Robert MilsonDept. Mathematics & StatisticsDalhousie [email protected]

David Gomez-UllateComplutense [email protected]

MS27

Doubling-up Hahn Polynomials: Classification andApplications

Joint work with J. Van der Jeugt. We examine how two sets

of Hahn polynomials Qn(x;α, β,N) and Qn(x; α, β, N) canbe combined into a single set of (discrete) orthogonal poly-

nomials Pn(x) (n = 0, 1, . . . , N+ N+1). Our analysis uses(new and old) shift operator relations for Hahn polynomi-als, coming from contiguous relations. This investigationgives rise to new classes of (rather pretty) tridiagonal ma-trices with a closed form spectrum, and has applications infinite quantum oscillators or in linear spin chains.

Roy Oste, Joris Van Der JeugtGhent [email protected], [email protected]

MS28

Overview of Digital Mathematics Libraries (DML)and the NIST Digital Repository of MathematicalFormulae (DRMF)

The DRMF is designed for a mathematically literate au-dience and should (1) facilitate interaction among mathe-maticians and scientists interested in compendia OPSF for-mulae data; (2) be expandable, allowing for input of newformulae from the literature; (3) use context-free semanticmarkup; (4) have a user friendly, consistent, and hyper-linkable viewpoint and authoring perspective; (5) performmath-aware search; and (6) use MathML for easily read,scalably rendered, content driven mathematics. A DMLoverview will be given.

Howard CohlApplied and Computational Mathematics DivisionNational Institute of Standards and [email protected]

MS28

Steps Toward Realizing a World Information Sys-tem for Digitally Organized Mathematics

The research mathematics available online has greatly in-creased over the last decade and a half; the methodsfor describing, linking and discovering these resources hasevolved much more slowly. A Global Digital Mathemat-ical Library Working Group was formed at the 2014 In-ternational Congress of Mathematicians. Concrete ideasfor creating a community-owned mathematical informationsystem for the digital age of science better supporting ad-vanced research in mathematics, and steps already takenwill be discussed.

Patrick IonAmerican Mathematical Society

[email protected]

MS28

Building DLMF

I will describe the technical processes and issues involved indeveloping and building the Digital Library of Mathemat-ical Functions (DLMF). These include: use of LATEX forauthoring for print; LATEXML to convert to web formatswith Presentation MathML; semantic markup for math-aware search and with an eye to future Content MathML.The applicability of these techniques to Digital Mathemat-ical Libraries and the web publishing of mathematics willbe addressed, as well as future work.


MS28

The Mathematica Computable Library of SpecialFunction Identities

We report on the enhancement and extension of the sub-stantial body of special function identities originally col-lected on the Wolfram Functions Site. A greatly aug-mented set of identities including a number of new func-tions and mathematical constants has now been integratedinto Wolfram|Alpha, making finding and working withthese identities easier than ever before. In addition, usingthe computable data framework pioneered for Mathemat-ica 10, this collection will soon be available in Mathematicaitself.

Eric WeissteinWolfram Research, [email protected]

MS29

Resurgence and Special Functions

The talk aims at presenting advances on some special func-tions from the viewpoint of resurgence theory. Amongother examples, the Painleve transcendents will be par-ticularly adressed.

Eric DelabaereUniversite [email protected]

MS29

Accelerating the Computation of Special FunctionsUsing Hybrid Hyper-Borel-Pade Approximations

In this talk we will unify the two current techniques of hy-perasymptotic expansions and Borel-Pade summation toaccelerate the high accuracy computation of special func-tions and their associated Stokes constants. Examples willbe given demonstrating the advantages, and disadvantages,of the techniques.

Chris HowlsSouthampton University

58 FA15 Abstracts

[email protected]

MS29

The Resurgence Properties of the Bessel and Han-kel Functions of Large Order and Argument

We reconsider the classical large order and argumentasymptotic series of the Bessel and Hankel functions dueto Debye. Employing the reformulation of the methodof steepest descents by Berry and Howls, we deriveresurgence-type formulas for the error terms of these ex-pansions. Using these new representations of the remain-der terms, we obtain numerically computable bounds andexponentially improved versions of Debye’s series, togetherwith asymptotic expansions for their late coefficients. Ouranalysis also provides a rigorous treatment of the formalresults derived earlier by Dingle.

Gergo NemesCentral European [email protected]

MS29

Exponentially Small Difference Between the Eigen-values am and bm+1 of Mathieu’s Equation

We consider Mathieu’s equation d2w(z)

dz2+(

λ− 2h2 cos 2z)w(z) = 0, which has special solutions

called Mathieu functions of integral order when λ = am orλ = bm+1, for m a positive integer. When h → ∞, thesespecial eigenvalues have the same asymptotic expansionsto all orders in descending powers of h. I will presentrigorous asymptotics to obtain the exponentially smalldifference between am and bm+1 as h→ ∞.

Karen OgilvieThe University of [email protected]

MS30

Recurrence Relations and Hamburger and StieltjesMoment Problems

Consider an orthogonal polynomial sequence (OPS){Pn(x)}∞n=0 which satisfies the three term recurrence re-lation

Pn(x) = (x− cn)Pn−1(x)− λnPn−2(x), cn real, λn+1 > 0, n = 1, 2, . . . ,

P0(x) = 1, P−1(x) = 0.

We are concerned with finding criteria which permit us todecide the determinacy or indeterminacy of the Hamburgerand/or Stieltjes moment problems associated with the re-currence relation above on the basis of the behavior of thetwo coefficient sequences in the recurrence relation. Theprototype for this type of result is the classical theorem ofCarleman [4]: The Hamburger moment problem associatedwith the recurrence relation above is determined if

∞∑n=2

λ−1/2n = ∞.

We will survey what has been done along these lines andpresent new results as well. We also recall a comparisontheorem due to Carleman which compares the coefficientsin a second three term recurrence relation with those in therecurrence relation above and concludes the determinacyor indeterminacy of the OPS determined by the second

recurrence relation from the determinacy or indeterminacyof the original OPS. We will also give new results of thissort.

Theodore ChiharaPurdue [email protected]

MS30

From Indeterminate to Determinate

In the talk, I’ll discuss various solutions to indeterminatemoment problems and explain how much (or little) oneneeds to modify the measure in order to change the natureof the problem. I’ll also discuss a number of conditionsthat usually won’t result in determinacy. This is mainlydone through examples and counterexamples.

Jacob S. ChristiansenUniversity of [email protected]

MS30

Spectral Properties of Unbounded Jacobi Matricesand Chihara’s Problem

Let μ be a determinate measure on the real line. Assumethe orthonormal polynomials {pn}∞n=0 with respect to μsatisfy

p−1(λ) ≡ 0, p0(λ) ≡ 1,

λpn(λ) = an−1pn−1(λ) + bnpn(λ) + anpn+1(λ) (n ≥ 0)

for sequences {an} and {bn}. The Chihara problem is thefollowing: assume

limn→∞

bn = ∞, limn→∞

a2nbnbn+1

=1

4

and the smallest accumulation point ρ of supp(μ) is fi-nite. Find additional conditions which imply supp(μ) ⊇[ρ,+∞). We are going to present a solution to this prob-lem.

Grzegorz SwiderskiInstytut Matematyczny, Uniwersytet [email protected]

MS31

Computer Algebra and Special Functions Inequal-ities

Proving special functions inequalities can be a tedious taskrequiring (a combination of) several different techniques.Among these techniques, algorithms for special functionsdeserve increasing attention. We will illustrate the scopeand limitations of existing computer algebra methods forproving special functions inequalities for some specific ex-amples that arose in some recent collaborations with GenoNikolov (et al.).

Veronika PillweinJohannes Kepler UniversityLInz, [email protected]

MS31

Inequalities and Bounds for Some Cumulative Dis-

FA15 Abstracts 59

tribution Functions

Bounds on some cumulative distributions functions andtheir inverses are discussed. In particular, we present re-cent bounds for incomplete gamma functions and Marcumfunctions (also called central and non-central cumulativegamma distributions) and we extend this type of resultsto other probability distributions like the beta distribution(central and noncentral). The inverse of the cumulativedistribution functions (quantile functions) are also impor-tant functions in statistics; we discuss how monotonicallyconvergent inversion methods can be used both for the nu-merical inversion and for obtaining analytical bounds forthese functions.

Javier SeguraUniversity of CantabriaSantander, [email protected]

MS31

Turan Type Inequalities for Struve Functions

We deduce some Turan type inequalities for Struve func-tions of the first kind by using various methods developedin the case of Bessel functions of the first and second kind.New formulas, like Mittag-Leffler expansion, infinite prod-uct representation for Struve functions of the first kind, areobtained, which may be of independent interest. More-over, some complete monotonicity results and functionalinequalities are deduced for Struve functions of the secondkind.

Sanjeev SinghIndian Institute of Technology [email protected]

Arpad BariczDepartment of Economics, Babes-Bolyai University,Cluj-Napoca 400591, [email protected]

Saminathan PonnusamyIndian Statistical Institute, Chennai Centre,CIT Campus, Taramani, Chennai 600113, [email protected]

MS32

Coupling Coefficients for Quantum SU(2) Repre-sentations

We study tensor products of infinite dimensional irre-ducible ∗-representations (not corepresentations) of theSU(2) quantum group. Eigenvectors of certain ’almost-central’ self-adjoint elements can be given explicitly interms of q-hypergeometric orthogonal polynomials. Wecompute coupling coefficients between different eigenvec-tors corresponding to the same eigenvalue; they turn outto be q-analogs of Bessel functions. As a result we obtainseveral q-integral identities involving q-hypergeometric or-thogonal polynomials and q-Bessel-type functions.

Wolter GroeneveltDelft Institute of Applied MathematicsTechnische Universiteit Delft

[email protected]

MS32

Ladder Operators for Rationally-Extended Poten-tials Connected with Exceptional Orthogonal Poly-nomials and Superintegrability

I will review results concerning k-step extension of the har-monic oscillator and the radial oscillator. These 1D ex-actly solvable systems are related to Hermite and Jacobiexceptional orthogonal polynomials of type III and allowdifferent types of ladder operators. I will show how lad-der operators involving no isolated multiplets exist andcan be constructed via combinations of Darboux-Crum andKrein-Adler SUSYQM approaches. I will also discuss theapplication to 2D superintegrable systems and derivationof their energy spectrum using polynomial algebras andtheir finite dimensional unitary representations. I will alsodiscuss how 1-step and 2-step extension of the harmonicoscillator are connected with a quantum Hamiltonian in-volving the fourth Painleve transcendent with third orderladder operators.

Ian MarquetteSchool of Mathematics and PhysicsThe University of [email protected]

MS32

BC1 Lame Polynomials

The potential of theBC1 quantum elliptic model is a super-position of two Weierstrass functions with doubling of bothperiods (two coupling constants). The BC1 elliptic modeldegenerates to A1 elliptic model characterized by the LameHamiltonian. It is shown that in the space of BC1 ellipticinvariant, the potential becomes a rational function, whilethe flat space metric becomes a polynomial. The modelpossesses the hidden sl(2) algebra for arbitrary couplingconstants: it is equivalent to sl(2)-quantum top in threedifferent magnetic fields. It is shown that there exist threeone-parametric families of coupling constants for which afinite number of polynomial eigenfunctions (up to a factor)occur. They can be called BC1 Lame polynomials, beinga generalization of the Lame polynomials.

Alexander TurbinerInstituto de Ciencias Nucleares, UNAMDepartamento de Gravitacion y Teora de [email protected]

MS32

A Dirac-Dunkl Equation on S2 and the Bannai-ItoAlgebra

The Dirac-Dunkl operator on the 2-sphere associated to theZ

32 reflection group is considered. Its symmetries are found

and are shown to generate the Bannai-Ito algebra. Repre-sentations of the Bannai-Ito algebra are constructed usingladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaiaextension theorem. These eigenfunctions, which corre-spond to Dunkl monogenics, are seen to support finite di-mensional irreducible representations of the BannaiIto al-gebra.

Hendrik De BieGhent [email protected]

60 FA15 Abstracts



MS33

An Overview of the Dynamic Dictionary of Mathe-matical Functions (http://ddmf.msr-inria.inria.fr)

DDMF is a generated, online, interactive dictionary of spe-cial functions. For each function, it displays:local andasymptotic expansions; recurrences on and closed forms ofthe coefficients in those expansions; guaranteed arbitrary-precision numerical approximations; plots; etc. More termsin expansions or more digits in approximations can beobtained upon request. When relevant, human-readableproofs are also automatically generated and displayed. Inthe talk, I will demonstrate the website and present thealgorithms used.

Frederic Chyzak

INRIA and Ecole Normale Superieure de [email protected]

MS33

MathSciNet: Digital Guide to the MathematicalLiterature

MathSciNet is the database based on Mathematical Re-views, a 75-year-old reviewing service covering the researchmathematics literature. With the exponential growth inscholarly publications, guides to the literature are increas-ingly important. There are several options available, rang-ing from simple internet searches to advanced databases.MathSciNet provides authoritative information about au-thors and publications, as well as abstracts and reviews ofmost of the literature. The talk includes a brief historyand discussions of scope and functionality.

Edward DunneAmerican Mathematical [email protected]

MS33

An Introduction to Recent Algorithms Behind theDDMF

The DDMF is built on the observation that many specialfunctions can be completely specified by a linear differen-tial equation and initial conditions. We will describe howthis data structure can be used to produce numerical valuesefficiently, expansions in Chebyshev series or other gener-alized Fourier expansions and also in some cases, continuedfraction expansions.

Bruno SalvyINRIA, [email protected]

MS33

Semantics, Formula Search, Mathematical Soft-ware - How the zbMATH Database extends Beyond

Publications

Condensing the corpus of mathematical research intoquickly accessible structured content has been the task ofreview services in mathematics for a long time. Tradition-ally done by humans (reviews, classification,...), the growthof research - increasingly dispersed in nonstandard formlike software - requires adapted scalable tools for contentanalysis. We introduce new automated tools, like math-aware taggers and semantic formula enrichment, currentlyemployed at zbMATH.

Olaf TeschkeZentralblatt [email protected]

MS34

Rigorous Borel Summability Methods and Appli-cations to Integrable Models

I will present some key results in rigorous Borel summabil-ity and their applications, with a special emphasis on twoproblems: the Dubrovin conjecture for Paineve P1 and amethods of calculating connection formulae in closed formin integrable models, without resorting to Riemann-Hilbertor similar reformulations, solely from the Painleve prop-erty.

Ovidiu CostinDepartment of MathematicsRutgers [email protected]

MS34

Uniform Asymptotics of Orthogonal PolynomialsArising from Coherent States

We study a family of orthogonal polynomials {φn(z)} aris-ing from nonlinear coherent states in quantum optics.Based on the three-term recurrence relation only, we ob-tain a uniform asymptotic expansion of φn(z) as the poly-nomial degree n tends to infinity. Our asymptotic resultssuggest that the weight function associated with the poly-nomials has an unusual singularity, which has never ap-peared for orthogonal polynomials in the Askey scheme.Our main technique is the Wang and Wong’s differenceequation method. In addition, the limiting zero distribu-tion of the polynomials φn(z) is provided.

Dan DaiCity University, Hong [email protected]

Weiying HuCity University of Hong [email protected]

Xiang-Sheng WangSoutheast Missouri State [email protected]

MS34

Some Recursive Techniques in the Approximationof Special Functions

Many special functions are solutions of a differential or dif-ference equation. An appropriate use of the Green functionof a certain part of the equation permits the transformation

FA15 Abstracts 61

of the differential or difference equation intro an integralor series equation respectively. Then, from the fixed pointtheorem of Banach we obtain a sequence of functions thatconverges to the given special function. As an illustration,we derive a new convergent expansion of the Bessel func-tions.

Jose L. LopezState University of [email protected]

Chelo Ferreira, Ester Perez SinusiaUniversidad de [email protected], [email protected]

MS34

Uniform Asymptotic Approximations for LinearDifferential Equations with a Bounded UniformityParameter

Typically when one studies uniform asymptotic approxi-mations for differential equations, the asymptotics is for alarge free parameter, say λ, and the approximations arevalid for the differentiation variable, say z, near a criticalpoint. Here we will discuss the opposite case. The differ-entiation variable is large, and the approximations are sup-posed to hold for the free parameter near a critical value.Note that in difference equations this is the typical situa-tion, since we normally study the large n asymptotics.

Adri B. Olde DaalhuisMaxwell Institute and School of MathematicsThe University of [email protected]

MS35

Generalizations of Generating Functions forMeixner and Krawtchouk Polynomials

In this talk we explain how connection and connection-typerelations for Meixner and Krawtchouk polynomials may bederived. Using these relations, we obtain generalizationsof generating functions for Meixner and Krawtchouk poly-nomials. From these generalized generating functions, wedevelop infinite series expressions using the discrete orthog-onality relations for Meixner and Krawtchouk polynomials.We will also explain how one may obtain orthogonality re-lations for these polynomials in the complex plane usingRamanujan’s master theorem.

Howard CohlNational Institute of Standards and TechnologyApplied and Computational Mathematics [email protected]

Roberto S. Costas-SantosDept. Physics and Mathematics Alcala de Henares,Madrid, [email protected]

Wenqing XuMontgomery Blair High [email protected]

MS35

Discrete Orthogonality of Four Types of Orthogo-

nal Polynomials of Weyl Groups

The link between the discrete Fourier calculus of the fourfamilies of special functions, C−, S−, Ss− and Sl− func-tions, and the four families of the induced orthogonal poly-nomials is discussed. The affine Weyl groups correspondingto the root systems of simple Lie algebras are recalled andsign homomorphisms, which allow general explicit descrip-tion of the orbit functions, are described. The discreteFourier calculus of the four types of orbit functions is per-formed for each type on a different set of points with theweights, labeling the orthogonal functions, chosen for eachtype separately. The four types of orthogonal polynomials,induced by the four types of orbit functions, inherit the dis-crete orthogonality from the orbit functions. The discreteorthogonality of the polynomials is explicitly formulatedand its application for the development of numerical inte-gration formulas and polynomial interpolation methods isdiscussed.

Jiri HrivnakDepartment of physicsCzech Technical University in [email protected]

MS35

Lattices of Any Dimension and Their Refinementto Any Density

A simple definition of a lattice in a real Euclidean spaceRn of dimension n, can be stated as follows: the infiniteset of discrete points Λ ∈ Rn is a lattice provided one hasΛ+Λ = Λ, where Λ+Λ stands for every sum of two pointsof Λ.Lattices in the two dimensional Euclidean plane are knownto be of two kinds, the lattice of squares and the lattice ofequilateral triangles. The lattices, characterized by theirsymmetries, come in two forms each. We identify the fourcases by the symbols most often used for the correspond-ing complex semisimple Lie algebras of rank 2: For squarelattices we use A1 × A1 and C2 , for triangular lattice weuse A2 and G2.

Marzena SzajewskaInstitute of Mathematics, University of [email protected]

MS35

Orthogonality of Macdonald Polynomials with Uni-tary Parameters

Generalizing previous work with Luc Vinet for the sym-metric group case, we show that for parameters q and t onthe unit circle and subject to a suitable truncation rela-tion, the Macdonald polynomials associated with crystal-lographic root systems satisfy a finite-dimensional systemof discrete orthogonality relations on the Weyl alcove. Thediscrete orthogonality weights are positive and their totalmass is expressed in product form by means of a finitelytruncated Aomoto-Ito type q-Selberg sum. This gives riseto a unitary q-deformed discrete Fourier involution on theWeyl alcove. For q = t our q-deformed Fourier transformamounts to the discrete Fourier transform associated withthe discrete orthogonality relations for the Weyl charactersstudied by Kirillov Jr., Korff and Stroppel, and also in amore general form by Patera et al. This is joint work withErdal Emsiz.

Jan Felipe van DiejenInstituto de Matematica y Fsica

62 FA15 Abstracts

Universidad de [email protected]

MS36

Precise and Fast Computation of Elliptic Functionsand Elliptic Integrals

New methods are developed to compute three Jaco-bian elliptic functions, complete and incomplete ellipticintegrals of all three kinds, and their derivatives andinversions by the half and double argument formulas.The new methods are of at least 50 bit accuracy and run1.1-3.5 times faster than the existing methods: Cody’sChebyshev approximations, Bulirsch’s cel and el1, andCarlson’s RF , RD, and RJ . All the published articlesand accompanied Fortran programs are available fromhttps://www.researchgate.net/profile/Toshio Fukushima/

Toshio FukushimaNational Astronomical Observatory of [email protected]

MS36

Computation and Inversion of Certain CumulativeDistribution Functions

Both the direct computation and the inversion of the cumu-lative central beta and gamma distributions (central andnoncentral) are used in many problems in statistics, ap-plied probability and engineering. Reliable and fast al-gorithms for the inversion and computation of these dis-tribution functions will be presented. Also, we will showcomparisons with other existing algorithms implementedin software platforms like Matlab, Mathematica and R. Inour algorithms, the computation of the cumulative distri-bution functions is based on the use of different methods ofapproximation such as Taylor expansions, continued frac-tions, uniform asymptotic expansions, numerical quadra-ture, etc depending on the parameter values. For the in-version, asymptotic expansions in combination with high-order Newton or secant methods are used.

Amparo GilDepto. de Matematicas, Est. y Comput.Universidad de [email protected]

Javier SeguraDepartamento de Matematicas, Estadistica yComputacionUniversidad de Cantabria, [email protected]

Nico M. TemmeCWI, [email protected]

MS36

On the Evaluation of Prolate Spheroidal WaveFunctions and Some Associated Quantities

Prolate spheroidal wave functions (PSWFs) correspondingto band limit c > 0 are the eigenfunctions of the truncatedFourier transform Fc : L2[−1, 1] → L2[−1, 1] defined viathe formula

Fc [σ] (x) =

∫ 1

−1

σ(t) · eicxt dt. (2)

In this capacity, PSWFs provide a natural tool for deal-ing with bandlimited functions defined on an interval, asdemonstrated by Slepian et. al. in a sequence of classicalpapers. (A function f : R → R is called bandlimited withband limit c > 0 if its Fourier transform is supported onthe interval [−c, c].) Starting with the papers by Slepian et.

al., PSWFs have been used as a tool in electrical engineer-ing (design of antenna patterns), digital signal processing(design of digital filters, such as upsampling/downsamplingalgorithms in acoustics), physics (various wave phenomena,fluid dynamics, uncertainty principles in quantum mechan-ics), etc. However, the use of PSWFs has been somewhat

crippled by their slightly mysterious reputation as being”difficult to compute”. This seems to be related to thefact that the classical (”Bouwkamp”) algorithm for theirevaluation encounters numerical difficulties for c > 40 orso. Moreover, the attempt to diagonalize the operator Fc

numerically via straightforward discretization meets withnumerical difficulties as well. In this talk, we describe sev-

eral numerical algorithms for the evaluation of PSWFs,some associated quantities, and PSWFs-based quadraturerules for the integration of bandlimited functions. Whilethe underlying analysis is somewhat involved, the resultingnumerical schemes are quite simple and efficient in practi-cal computations, even for large values of band limit (e.g.c = 106).

Andrei OsipovMathematics DepartmentYale [email protected]

MS36

A Fast Chebyshev-Legendre Transform Using AnAsymptotic Formula

Legendre expansions have applications throughout scien-tific computing because of their L2-orthogonality, rapidlydecaying Cauchy transform, and association with spher-ical harmonics. However, fast algorithms for comput-ing with Legendre expansions are not readily available.In this talk we describe a fast and numerically stableO(N(logN)2/ log logN) algorithm for converting betweenLegendre and Chebyshev expansions based on carefully ex-ploiting an asymptotic formula. Applications are: fast L2-projection, mollification, and the effective computation ofPainleve transcendents.

Alex TownsendDepartment of [email protected]

Nick HaleStellenbosch [email protected]

MS37

The First Szego’s Limit Theorem with Varying Co-efficients

In this talk, I will present some extensions of Szego’s FirstLimit Theorem (SFLT) to a class of non Toeplitz matrices.Our results extend those of Kuijlaars and Van Assche forJacobi matrices, as well as those of Tilli on locally Toeplitz

FA15 Abstracts 63

matrices.

Alain BourgetCal State University, FullertonDepartment of [email protected]

MS37

The Second Szego’s Limit Theorem for a Class ofnon-Toeplitz Matrices

Let a(t) be an integrable function on the unit circle withFourier coefficients

ak =1

2π

∫ π

−π

a(t) e−ikt dt, k ∈ Z.

The Toeplitz matrices Tn(a) associated with a are

Tn(a) = {aj−k}0≤j,k≤n−1 .

Szego’s First Limit Theorem (SFLT) states that for anyself-adjoint Toeplitz operator T (a) and any continuous ϕ,

Tr[ϕ(Tn(a))] =n

2π

∫ π

−π

ϕ(a(t)) dt+ o(n).

Almost 40 years after establishing the FSLT, Szego estab-lished the second Szego’s limit theorem which gives a for-mula for the o(n) error term above up to exponentiallysmall terms. The SFLT has recently been extended to aclass of matrices whose diagonal entries satisfy a small de-viation condition. In this talk I will present results towardextending the second limit theorem to this class of non-Toeplitz matrices.

Tyler McMillenCalifornia State University, FullertonDepartment of [email protected]

MS37

Ratio Asymptotics and Weak Asymptotic Mea-sures

We will consider the asymptotics of general orthogonalpolynomials in the complex plane. Our focus will be onratio asymptotics and weak asymptotic measures and howthey relate to the right limits of the Bergman Shift matrix.We will also discuss relative forms of these asymptoticsand generalize some results from the setting of OPUC tomeasures with more general support.

Brian SimanekVanderbilt [email protected]

MS37

Newman-Rivlin Asymptotics for Partial Sums ofPower Series

We discuss analogues of Newman and Rivlin’s formula con-cerning the ratio of a partial sum of a power series to itslimit function and present a new general result of this typefor entire functions with a certain asymptotic character.The main tool used in the proof is a Riemann-Hilbert for-mulation for the partial sums introduced by Kriecherbaueret al. This new result makes some progress on verifying

a part of the Saff-Varga Width Conjecture concerning thezero-free regions of these partial sums.

Antonio R. VargasDalhousie UniversityDepartment of Mathematics and [email protected]

MS38

Universality of Mesoscopic Fluctuations in Orthog-onal Polynomial Ensembles

We shall discuss fluctuations on the mesoscopic scale fororthogonal polynomial ensembles and show that these areuniversal in the sense that two measures with asymptoticrecurrence coefficients have the same asymptotic meso-scopic fluctuations (under an additional assumption on thelocal regularity of one of the measures). The convergencerate of the recurrence coefficients determines the range ofscales on which the limiting fluctuations are identical. Aparticular consequence of our results is a Central LimitTheorem for the modified Jacobi Unitary Ensembles on allmesoscopic scales. This is joint work with Maurice Duits.

Jonathan BreuerHebrew University of Jerusalem, [email protected]

MS38

Orthogonal Dirichlet Polynomials

We discuss orthogonal ”polynomials” formed from linearcombinations of Dirichlet functions, as used in L series innumber theory. Although formed from the same basis func-tions as used in the continuous orthogonality of Krein sys-tems, they are distinct from Krein systems. We motivatetheir study, and present some recent results.

Doron S. LubinskyGeorgia Institute of [email protected]

MS38

Analytic Continuation of S-property for MultipleOrthogonal Polynomials

We are interested in the S-property for multiple orthogonalpolynomials. As a case study, we analyze a family of mul-tiple orthogonal polynomials arising in the normal matrixmodel with cubic + linear potential. In order to performanalytic continuation of the S-property, we interpret it interms of a quadratic differential on the associated spectralcurve, and develop a deformation argument on its criticalgraph.

Guilherme SilvaKatholieke Universiteit Leuven, [email protected]

Pavel BleherDepartment of Mathematics, [email protected]

MS38

Nuttall’s Theorem on Algebraic S-contours

Given a function f holomorphic at infinity, the n-th diago-nal Pade approximant to f , say [n/n]f , is a rational func-

64 FA15 Abstracts

tion of type (n, n) that has the highest order of contactwith f at infinity. Nuttall’s theorem provides an asymp-totic formula for the error of approximation f − [n/n]f inthe case where f is the Cauchy integral of a smooth den-sity with respect to the arcsine distribution on [−1, 1]. Iwill present an extension of Nuttall’s theorem to Cauchyintegrals on the so-called algebraic S-contours.

Maxim YattselevIndiana University-Purdue University [email protected]

MS39

Geometry of Hermite-Pade Approximants for aPair of Cauchy Transforms with Interlacing Sym-metric Supports

We consider the multiple orthogonal polynomials with re-spect to smooth complex measures supported on the realline. In this talk we are interested in the case where thesupports form two interlacing symmetric intervals and theratio of the measures extends to a holomorphic functionin a region that depends on the size of interlacing. Thisproblem was posed and studied by Herbert Stahl in the80’s. We shall speak about algebraic functions (of genus 1and 2) and their abelian integrals (with purely imaginaryperiods) which define the main term of the asymptotics forthis problem. This is joint work with Walter Van Assche

and Maxim L. Yattselev.

Alexander I. AptekarevKeldysh Institute of Applied MathematicsRussian Academy of [email protected]

MS39

Difference Operators on Lattices and Multiple Or-thogonal Polynomials

We construct difference operators on Z2 using multiple or-thogonal polynomials. Let us stress that it is not clearwhether the eigenvalue problem for a difference equation onZ2 has a solution and, especially, whether the entries of aneigenvector can be chosen to be polynomials in the spectralvariable. However, for the difference operators in question,the existence of a polynomial solution to the eigenvalueproblem is guaranteed if the coefficients of the differenceoperators satisfy a certain discrete zero curvature condi-tion. In turn, this means that there is a discrete integrablesystem behind the scene and the discrete integrable systemcan be thought of as a generalization of what is known asthe discrete time Toda equation, which appeared for thefirst time as the Frobenius identity for the elements of thePade table.

Maxim DerevyaginUniversity of [email protected]

MS39

Weak and Strong Asymptotics for the PollaczekMultiple Orthogonal Polynomials

Pollaczek multiple orthogonal polynomials are type IIHermite–Pade polynomials orthogonal with respect to twosimple measures supported on the positive semi–axis.These measures form a socalled Nikishin pair, with the

feature that one of its generators is purely discrete. It isknown that the largedegree asymptotics of such polyno-mials is governed by the solution of a vector equilibriumproblem, which was previously computed by V. Sorokin.For the strong asymptotics we use the Riemann–Hilbertcharacterization of the Hermite–Pade polynomials and thecorresponding nonlinear steepest descent method. We dis-cuss some of the main ingredients of this analysis and theasymptotic results obtained by this method. This is a jointwork with A. Aptekarev and G. Lopez-Lagomasino

Andrei Martinez-FinkelshteinUniversity of [email protected]

MS39

On the Convergence of Mixed Type Hermite-PadeApproximants

The convergence of diagonal sequences of type II Hermite-Pade approximants of Nikishin systems have been knownfor some time and recently similar results have been ob-tained for type I Hermite-Pade approximants. In this talkwe present new results on the convergence of diagonal se-quences of a certain mixed type Hermite-Pade approxi-mation problem of a Nikishin system, which is motivatedin finding approximating solutions of a Degasperis-Procesipeakons problem and in the study of the inverse spectralproblem for the discrete cubic string.

Sergio M. Medina PeraltaUniversidad Carlos III de [email protected]

Guillermo Lopez [email protected]

MS40

Singular Linear Statistics of the Laguerre UnitaryEnsemble and Painleve III: Double Scaling Analy-sis

We compute the Hankel determinant generated from asingularly deformed weight, w(x; t, α) := xα exp(−x −t/x), x > 0, t > 0, in a double scaling scheme, namely,n → ∞ and t → 0, such that s = (2n + 1 + α)t is fi-nite. Asymptotic expansions of the double-scaled determi-nant are obtained for large and small s. These are foundthrough solutions of a particular Painleve III obtained fromthe original finite n PIII (with a larger number of param-eters.)

Yang ChenUniversity of [email protected]

MS40

Christoffel-Darboux-type Formulae for Orthonor-mal Rational Functions and Asymptotics

Consider the reproducing kernel Kn(x, y) =∑n−1k=0 ϕk(x)ϕk(y), where {ϕk}n−1

k=0 forms an orthonormalbasis for the space of rational functions with poles among{α1, α2, . . . , αn−1} ⊂ C ∪ {∞}. In the first part of thetalk we present Christoffel-Darboux-type formulae for Kn

by exploiting its relation with so-called quasi-orthogonalrational functions. In the second part we use these

FA15 Abstracts 65

formulae to obtain asymptotics for the reproducing kernelKn for n→ ∞.

Karl DeckersUniversity of Lille, [email protected]

MS40

Recent Asymptotic Expansions for Legendre Poly-nomial Expansions and Gauss-Legendre Quadra-ture

In this talk, we discuss some recent asymptotic expansionsrelated to problems in approximation theory and numericalquadrature.

• We present a full asymptotic expansion, as n → ∞,of the Legendre polynomial Pn(x) for |x| ≤ 1 − ε,ε ∈ (0, 1).

• We present full asymptotic expansions, as n →∞, of Legendre series coefficients an = (n +

1/2)∫ 1

−1f(x)Pn(x)dx, when f(x) has arbitrary

algebraic-logarithmic (interior and/or endpoint) sin-gularities in [−1, 1].

• We present a full asymptotic expansion (as thenumber of abscissas tends to infinity) for Gauss–Legendre quadrature formula

∑ni=1 wnif(xni) for in-

tegrals∫ 1

−1f(x)dx, where f(x) is allowed to have ar-

bitrary algebraic-logarithmic endpoint singularities.

Avram SidiComputer Science DepartmentTechnion - Israel Institute of [email protected]

FA15 Abstracts - SIAM · 2018. 5. 25. · 34 FA15 Abstracts NataliaC.Pinz´on-Cort´es...

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