Quasi-definiteness of generalized Uvarov transforms of moment … · 2019. 8. 1. · 70 Generalized...

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QUASI-DEFINITENESS OF GENERALIZED UVAROV TRANSFORMS OF MOMENT FUNCTIONALS D. H. KIM AND K. H. KWON Received 11 March 2001 When σ is a quasi-definite moment functional with the monic orthogonal polynomial system {P n (x)} n=0 , we consider a point masses perturbation τ of σ given by τ := σ + λ m l=1 m l k=0 ((−1) k u lk /k!)δ (k) (x c l ), where λ, u lk , and c l are constants with c i = c j for i = j. That is, τ is a gen- eralized Uvarov transform of σ satisfying A(x)τ = A(x)σ, where A(x)= m l=1 (x c l ) m l +1 . We find necessary and sufficient conditions for τ to be quasi-definite. We also discuss various properties of monic orthogonal poly- nomial system {R n (x)} n=0 relative to τ including two examples. 1. Introduction In the study of Pade ´ approximation (see [5, 10, 21]) of Stieltjes type mero- morphic functions b a (x) z x + m l=1 m l k=0 C lk k! ( z c l ) k+1 , (1.1) where a<b , C lk are constants, and (x) is a positive Stieltjes measure, the denominators R n (x) of the main diagonal sequence of Pade ´ approximants satisfy the orthogonality b a R n (x)π(x)(x)+ m l=1 m l k=0 C lk k πR n ( c l ) = 0, π P n1 , (1.2) Copyright c 2001 Hindawi Publishing Corporation Journal of Applied Mathematics 1:2 (2001) 69–90 2000 Mathematics Subject Classification: 33C45 URL: http://jam.hindawi.com/volume-1/S1110757X01000225.html

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Page 1: Quasi-definiteness of generalized Uvarov transforms of moment … · 2019. 8. 1. · 70 Generalized Uvarov transforms where P n is the space of polynomials of degree ≤ n with P−1

QUASI-DEFINITENESS OF GENERALIZED UVAROVTRANSFORMS OF MOMENT FUNCTIONALS

D. H. KIM AND K. H. KWON

Received 11 March 2001

When σ is a quasi-definite moment functional with the monic orthogonalpolynomial system {Pn(x)}∞n=0, we consider a point masses perturbation τ

of σ given by τ := σ + λ∑m

l=1

∑ml

k=0((−1)kulk/k!)δ(k)(x − cl), where λ,

ulk, and cl are constants with ci �= cj for i �= j. That is, τ is a gen-eralized Uvarov transform of σ satisfying A(x)τ = A(x)σ, where A(x) =∏m

l=1(x− cl)ml+1. We find necessary and sufficient conditions for τ to be

quasi-definite. We also discuss various properties of monic orthogonal poly-nomial system {Rn(x)}∞n=0 relative to τ including two examples.

1. Introduction

In the study of Pade approximation (see [5, 10, 21]) of Stieltjes type mero-morphic functions

∫b

a

dµ(x)

z−x+

m∑l=1

ml∑k=0

Clkk!(

z−cl

)k+1, (1.1)

where −∞ ≤ a < b ≤ ∞, Clk are constants, and dµ(x) is a positive Stieltjesmeasure, the denominators Rn(x) of the main diagonal sequence of Padeapproximants satisfy the orthogonality

∫b

a

Rn(x)π(x)dµ(x)+

m∑l=1

ml∑k=0

Clk∂k[πRn

](cl

)= 0, π ∈ Pn−1, (1.2)

Copyright c© 2001 Hindawi Publishing CorporationJournal of Applied Mathematics 1:2 (2001) 69–902000 Mathematics Subject Classification: 33C45URL: http://jam.hindawi.com/volume-1/S1110757X01000225.html

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70 Generalized Uvarov transforms

where Pn is the space of polynomials of degree ≤ n with P−1 = {0}. That is,Rn(x) (n ≥ 0) are orthogonal with respect to the measure

dµ+

m∑l=1

ml∑k=0

(−1)kClkδ(k)(x−cl

), (1.3)

which is a point masses perturbation of dµ(x). Orthogonality to a positiveor signed measure perturbed by one or two point masses arises naturallyalso in orthogonal polynomial eigenfunctions of higher order (≥ 4) ordi-nary differential equations (see [14, 15, 16, 19]), which generalize Bochner’sclassification of classical orthogonal polynomials (see [6, 18]). On the otherhand, many authors have studied various aspects of orthogonal polynomialswith respect to various point masses perturbations of positive-definite (see[1, 2, 8, 14, 27, 28]) and quasi-definite (see [3, 4, 9, 11, 12, 20, 23]) momentfunctionals. In this work, we consider the most general such situation. Thatis, we consider a moment functional τ given by

τ := σ+λ

m∑l=1

ml∑k=0

(−1)kulk

k!δ(k)

(x−cl

), (1.4)

where σ is a given quasi-definite moment functional, λ, ulk, and cl arecomplex numbers with ul,ml

�= 0 and ci �= cj for i �= j, that is, τ is obtainedfrom σ by adding a distribution with finite support. We give necessary andsufficient conditions for τ to be quasi-definite. When τ is also quasi-definite,we discuss various properties of orthogonal polynomials {Rn(x)}∞n=0 relativeto τ in connection with orthogonal polynomials {Pn(x)}∞n=0 relative to σ.These generalize many previous works in [4, 9, 11, 12, 20, 23].

2. Necessary and sufficient conditions

For any integer n ≥ 0, let Pn be the space of polynomials of degree ≤ n andP =

⋃n≥0 Pn. For any π(x) in P, let deg(π) be the degree of π(x) with the

convention that deg(0) = −1. For the moment functionals σ, τ (i.e., linearfunctionals on P) (see [7]), c in C, and a polynomial φ(x) =

∑nk=0 akxk, let

⟨σ′,π

⟩:= −

⟨σ,π′⟩; 〈φσ,π〉 := 〈σ,φπ〉;⟨

(x−c)−1σ,π⟩

:=⟨σ,θcπ

⟩;

(θcπ

)(x) :=

π(x)−π(c)

x−c; (2.1)

(σφ)(x) :=

n∑k=0

(n∑

j=k

ajσjk

)xk; 〈στ,π〉 = 〈σ,τπ〉, π ∈ P.

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D. H. Kim and K. H. Kwon 71

We also let

F(σ)(z) :=

∞∑n=0

σn

zn+1(2.2)

be the (formal) Stieltjes function of σ, where σn := 〈σ,xn〉 (n ≥ 0) are themoments of σ. Following Zhedanov [29], for any polynomials A(z), B(z),

C(z), D(z) with no common zero and |C|+ |D| �≡ 0, let

S(A,B,C,D)F(σ)(z) :=AF(σ)+B

CF(σ)+D. (2.3)

If S(A,B,C,D)F(σ) = F(τ) for some moment functional τ, then we call τ arational (resp., linear) spectral transform of σ (resp., when C(z) = 0). ThenS(A,B,C,D)F(σ) = F(τ) if and only if

xA(x)σ = C(x)(στ)+xD(x)τ,

〈σ,A〉+x(σθ0A

)(x)+xB(x) = (στ)

(θ0C

)(x)+〈τ,D〉+x

(τθ0D

)(x).

(2.4)

In particular, for any c and β in C, let

U(c,β)F(σ) :=(z−c)F(σ)+β

z−c(2.5)

be the Uvarov transform (see [28, 29]) of F(σ). Then for any {ci}ki=1 and

{βi}ki=1 in C,

F(τ) := U(ck,βk

) · · ·U(c1,β1

)F(σ) =

A(z)F(σ)+B(z)

A(z), (2.6)

where A(z) =∏k

i=1(z−ci), B(z) =∑k

i=1 βi

∑kj=1j �=i

(z−cj), and by (2.4)

A(x)τ = A(x)σ. (2.7)

In this case, we say that τ is a generalized Uvarov transform of σ. Conversely,if (2.7) holds for some polynomial A(x) (�≡ 0), then

F(τ) =A(z)F(σ)+

(τθ0A

)(z)−

(σθ0A

)(z)

A(z)(2.8)

and F(τ) is obtained from F(σ) by deg(A) successive Uvarov transforms (see[29]), that is, τ is a generalized Uvarov transform of σ.

In the following, we always assume that τ is a moment functional givenby (1.4), where σ is a quasi-definite moment functional. Let {Pn(x)}∞n=0 bethe monic orthogonal polynomial system (MOPS) relative to σ satisfying the

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72 Generalized Uvarov transforms

three term recurrence relation

Pn+1(x) =(x−bn

)Pn(x)−cnPn−1(x), n ≥ 0,

(P−1(x) = 0

). (2.9)

Since (1.4) implies (2.7) with A(x) =∏m

l=1(x−cl)ml+1, τ is a generalized

Uvarov transform of σ. Then our main concern is to find conditions underwhich a generalized Uvarov transform τ, given by (1.4), of σ is also quasi-definite. In other words, we are to solve the division problem (2.7) of themoment functionals.

Let

Kn(x,y) :=

n∑j=0

Pj(x)Pj(y)⟨σ,P2

j

⟩ , n ≥ 0 (2.10)

be the nth kernel polynomial for {Pn(x)}∞n=0 and K(i,j)n (x,y)=∂x

i∂yjKn(x,y).

We need the following lemma which is easy to prove.

Lemma 2.1. Let V = (x1,x2, . . . ,xn)t and W = (y1,y2, . . . ,yn)t be two vec-tors in C

n. Then

det(In +VWt

)= 1+

n∑j=1

xjyj, n ≥ 1, (2.11)

where In is the n×n identity matrix.

Theorem 2.2. The moment functional τ is quasi-definite if and only ifdn �= 0, n ≥ 0, where dn is the determinant of (

∑ml=1(ml + 1)) ×

(∑m

l=1(ml +1)) matrix Dn:

Dn :=[Atl(n)

]m

t,l=1, n ≥ 0, (2.12)

where

Atl(n) =

[δtlδki +λ

ml−i∑j=0

ul,i+j

i!j!K(k,j)

n

(ct,cl

)]mt ml

k=0, i=0

. (2.13)

If τ is quasi-definite, then the MOPS {Rn(x)}∞n=0 relative to τ is given by

Rn(x) = Pn(x)−λ

m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!K

(0,j)n−1

(x,cl

)R(i)

n

(cl

), (2.14)

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D. H. Kim and K. H. Kwon 73

where {R(i)n (cl)}

m ml

l=1,i=0 are given by

Dn−1

Rn

(c1

)R′

n

(c1

)...

R(m1)n

(c1

)Rn

(c2

)...

R(mm)n

(cm

)

=

Pn

(c1

)P′

n

(c1

)...

P(m1)n

(c1

)Pn

(c2

)...

P(mm)n

(cm

)

, n ≥ 0(D−1 = I

). (2.15)

Moreover, ⟨τ,R2

n

⟩=

dn

dn−1

⟨σ,P2

n

⟩, n ≥ 0

(d−1 = 1

). (2.16)

Proof. (⇒). Assume that τ is quasi-definite and expand Rn(x) as

Rn(x) = Pn(x)+

n−1∑j=0

CnjPj(x), n ≥ 1, (2.17)

where Cnj = 〈σ,RnPj〉/〈σ,P2j 〉, with 0 ≤ j ≤ n−1. Here,

⟨σ,RnPj

⟩=

⟨τ−λ

m∑l=1

ml∑k=0

(−1)kulk

k!δ(k)

(x−cl

),RnPj

= −λ

m∑l=1

ml∑k=0

ulk

k!

k∑i=0

(k

i

)R(i)

n

(cl

)P

(k−i)j

(cl

) (2.18)

so that

Rn(x) = Pn(x)−λ

n−1∑j=0

Pj(x)⟨σ,P2

j

⟩ m∑l=1

ml∑k=0

ulk

k!

k∑i=0

(k

i

)R(i)

n

(cl

)P

(k−i)j

(cl

)

= Pn(x)−λ

m∑l=1

ml∑k=0

ulk

k!

k∑i=0

(k

i

)R(i)

n

(cl

)K

(0,k−i)n−1

(x,cl

)

= Pn(x)−λ

m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!K

(0,j)n−1

(x,cl

)R(i)

n

(cl

).

(2.19)

Hence, we have (2.14). Set the matrices Bl and El to be

Bl =

Rn

(cl

)R

′n

(cl

)...

R(ml)n

(cl

)

, El =

Pn

(cl

)P

′n

(cl

)...

P(ml)n

(cl

)

, 1 ≤ l ≤ m. (2.20)

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74 Generalized Uvarov transforms

Then,

E1

E2

...Em

=

[Atl(n−1)

]m

t,l=1

B1

B2

...Bm

, (2.21)

which gives (2.15). Now,

Dn =[Atl(n)

]m

t,l=1

= Dn−1 +λ⟨

σ,P2n

⟩[[

ml−i∑j=0

ul,i+j

i!j!P(j)

n

(cl

)P(k)

n

(ct

)]mt ml

k=0, i=0

]m

t,l=1

= Dn−1 +λ⟨

σ,P2n

E1

E2

...Em

[m1∑j=0

u1j

j!P(j)

n

(c1

),

m1−1∑j=0

u1,j+1

j!P(j)

n

(c1

), . . . ,

u1,m1

m1!Pn

(c1

),

m2∑j=0

u2j

j!P(j)

n

(c2

), . . . ,

um,mm

mm!Pn

(cm

)]

= Dn−1

I+

λ⟨σ,P2

n

B1

B2

...Bm

[m1∑j=0

u1j

j!P(j)

n

(c1

),

m1−1∑j=0

u1,j+1

j!P(j)

n

(c1

), . . . ,

u1,m1

m1!Pn

(c1

),

m2∑j=0

u2j

j!P(j)

n

(c2

), . . . ,

um,mm

mm!Pn

(cm

)] (2.22)

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D. H. Kim and K. H. Kwon 75

so that

dn = dn−1

(1+

λ⟨σ,P2

n

⟩ m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!P(j)

n

(cl

)R(i)

n

(cl

))(2.23)

by Lemma 2.1. On the other hand,

⟨τ,R2

n

⟩=

⟨τ,RnPn

⟩=

⟨σ,RnPn

⟩+λ

⟨m∑

l=1

ml∑k=0

(−1)kulk

k!δ(k)

(x−cl

),RnPn

=⟨σ,P2

n

⟩+λ

m∑l=1

ml∑k=0

ulk

k!

k∑j=0

(k

j

)R(j)

n

(cl

)P(k−j)

n

(cl

)

=⟨σ,P2

n

⟩+λ

m∑l=1

ml∑j=0

ml∑k=j

ulk

k!

(k

j

)R(j)

n

(cl

)P(k−j)

n

(cl

)

(2.24)

so that

⟨τ,R2

n

⟩=

⟨σ,P2

n

⟩+λ

m∑l=1

ml∑j=0

ml−j∑k=0

ul,j+k

j!k!R(j)

n

(cl

)P(k)

n

(cl

). (2.25)

Hence, from (2.23) and (2.25), we have

⟨σ,P2

n

⟩dn = dn−1

⟨τ,R2

n

⟩, n ≥ 0. (2.26)

Note that (2.26) also holds for n = 0 if we take d−1 = 1. Hence, dn �= 0,

n ≥ 0 inductively and we have (2.16).(⇐). Assume that dn �= 0, with n ≥ 0 and define {Rn(x)}∞n=0 by (2.14).

Then we have, by (2.14) and (2.23),

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76 Generalized Uvarov transforms

⟨τ,RnPr

⟩=

⟨σ,RnPr

⟩+λ

⟨m∑

l=1

ml∑k=0

(−1)kulk

k!δ(k)

(x−cl

),RnPr

=⟨σ,RnPr

⟩+λ

m∑l=1

ml∑k=0

ulk

k!

k∑j=0

(k

j

)R(j)

n

(cl

)P(k−j)

r

(cl

)

=⟨σ,PnPr

⟩−λ

m∑l=1

ml∑j=0

ml−j∑k=0

ul,j+k

j!k!R(j)

n

(cl

)⟨σ,K

(0,k)n−1

(x,cl

)Pr(x)

m∑l=1

ml∑j=0

ml−j∑k=0

ul,j+k

j!k!R(j)

n

(cl

)P(k)

r

(cl

)

=⟨σ,PnPr

⟩−λ

m∑l=1

ml∑j=0

ml−j∑k=0

ul,j+k

j!k!R(j)

n

(cl

)P(k)

r

(cl

)(1−δnr

)

m∑l=1

ml∑j=0

ml−j∑k=0

ul,j+k

j!k!R(j)

n

(cl

)P(k)

r

(cl

)

=

0, 0 ≤ r ≤ n−1,

⟨σ,P2

n

⟩+λ

m∑l=1

ml∑j=0

ml−j∑k=0

ul,j+k

j!k!P(k)

n

(cl

)R(j)

n

(cl

), r = n,

=

0, 0 ≤ r ≤ n−1,

dn

dn−1

⟨σ,P2

n

⟩ �= 0, r = n,

(2.27)

since 〈σ,K(0,k)n−1 (x,cl)Pr(x)〉 = P

(k)r (cl)(1−δnr). Hence,

⟨τ,RnRm

⟩=

{0, if 0 ≤ m ≤ n−1,⟨τ,RnPn

⟩ �= 0, if m = n,(2.28)

so that {Rn(x)}∞n=0 is the MOPS relative to τ and so τ is also quasi-definite.�

General division problems of moment functionals

D(x)τ = A(x)σ (2.29)

is handled in [17], when D(x) and A(x) have no common zero. Theorem 2.2includes the following as special cases.

• m = 1, m1 = 0: Marcellan and Maroni [23],• m = 2, m1 = m2 = 0: Draıdi and Maroni [9], Kwon and Park [20],

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D. H. Kim and K. H. Kwon 77

• m = 1, m1 = 1: Belmehdi and Marcellan [4],• m = 1: Kim, Kwon, and Park [12].

Some other special cases where σ is a classical moment functional werehandled in [2, 1, 3, 8, 11, 14].

From now on, we always assume that dn �= 0, with n ≥ 0, so that τ is alsoquasi-definite.

Theorem 2.3. For the MOPS {Rn(x)}∞n=0 relative to τ, we have(i) (the three-term recurrence relation)

Rn+1(x) =(x−βn

)Rn(x)−γnRn−1(x), n ≥ 0, (2.30)

where

βn = bn +λ⟨

σ,P2n

⟩ m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!

×{P(j)

n

(cl

)R

(i)n+1

(cl

)−P

(j)n−1

(cl

)R(i)

n

(cl

)}(n ≥ 0),

(2.31)

γn =dndn−2

d2n−1

cn (n ≥ 1). (2.32)

(ii) (the quasi-orthogonality)

m∏l=1

(x−cl

)ml+1Rn(x) =

n+r∑j=n−r

CnjPj(x), n ≥ r, (2.33)

where r =∑m

l=1(ml +1), Cn,n−r �= 0, and

Cnj =

⟨σ,

∏ml=1

(x−cl

)ml+1RnPj

⟩⟨σ,P2

j

⟩=

⟨τ,

∏ml=1

(x−cl

)ml+1RnPj

⟩⟨σ,P2

j

⟩ , where n−r ≤ j ≤ n+r.

(2.34)

Proof. For (i), by (2.14), we can rewrite (2.30) as

Pn+1(x)−λ

m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!K(0,j)

n

(x,cl

)R

(i)n+1

(cl

)

=(x−βn

){Pn(x)−λ

m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!K

(0,j)n−1

(x,cl

)R(i)

n

(cl

)}

−γn

{Pn−1(x)−λ

m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!K

(0,j)n−2

(x,cl

)R

(i)n−1

(cl

)}.

(2.35)

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78 Generalized Uvarov transforms

After multiplying (2.35) by Pn(x) and applying σ, we have

−λ

m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!P(j)

n

(cl

)R

(i)n+1

(cl

)

=(bn −βn

)⟨σ,P2

n

⟩−λ

m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!P

(j)n−1

(cl

)R(i)

n

(cl

).

(2.36)

Hence, we have (2.31) and by (2.16)

γn =

⟨τ,R2

n

⟩⟨τ,R2

n−1

⟩ =dndn−2

d2n−1

cn (n ≥ 1). (2.37)

For (ii),∏m

l=1(x−cl)ml+1Rn(x)=

∑n+rj=0 CnjPj(x), where r=

∑ml=1(ml+1)

and

Cnk

⟨σ,P2

k

⟩=

⟨σ,

m∏l=1

(x−cl

)ml+1Rn(x)Pk(x)

=

⟨m∏

l=1

(x−cl

)ml+1τ,Rn(x)Pk(x)

=

⟨τ,

m∏l=1

(x−cl

)ml+1Rn(x)Pk(x)

⟩= 0, if r+k < n.

(2.38)

Hence, Cnk = 0, 0 ≤ k ≤ n− r− 1 and Cn,n−r �= 0 so that we have (2.33)and (2.34). �

Corollary 2.4. Assume that σ is positive-definite and let [ξ,η] be the trueinterval of the orthogonality of σ. Then

(i)∏m

l=1(x−cl)ml+1Rn(x) has at least n−r distinct nodal zeros (i.e.,

zeros of odd multiplicities) in (ξ,η).(ii) Rn(x) has at least n−r−m distinct nodal zeros in (ξ,η).If furthermore ml (1 ≤ l ≤ m) are odd or ξ ≥ cl (1 ≤ l ≤ m), then(iii) Rn(x) has at least n−r distinct nodal zeros in (ξ,η).

Proof. (i) and (ii) are trivial by (2.33).For (iii), assume that ml (1 ≤ l ≤ m) are odd. Then σ :=

∏ml=1(x −

cl)ml+1σ is also positive-definite on [ξ,η]. Let {Pn(x)}∞n=0 be the MOPS

relative to σ. Then we may write Rn(x) =∑n

j=0 CnjPj(x), where

Cnk

⟨σ, P2

k

⟩=

⟨σ,RnPk

⟩=

⟨m∏

l=1

(x−cl

)ml+1τ,RnPk

=

⟨τ,

m∏l=1

(x−cl

)ml+1RnPk

⟩.

(2.39)

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D. H. Kim and K. H. Kwon 79

Hence, Cnk = 0, 0 ≤ k ≤ n−r−1 so that Rn(x) =∑n

j=n−r CnjPj(x). Hence,Rn(x) has at least n− r distinct nodal zeros in (ξ,η). In case ξ ≥ cl (1 ≤l ≤ m), σ =

∏ml=1(x−cl)

ml+1σ is also positive-definite on [ξ,η] so that bythe same reasoning as above Rn(x) has at least n − r distinct nodal zerosin (ξ,η). �

Theorem 2.5. For any polynomial p(x) of degree at most n, we have⟨τ,L(0,k)

n (x,y)p(x)⟩

= p(k)(y), (2.40)

where Ln(x,y) =∑n

i=0 Ri(x)Ri(y)/〈τ,R2i 〉, n ≥ 0, is the nth kernel poly-

nomial for {Rn(x)}∞n=0 and

Ln(x,y) = Kn(x,y)−λ

dn

m∑l=1

ml∑i=0

∣∣Dun

∣∣ml−i∑j=0

ul,i+j

i!j!K(0,j)

n

(x,cl

), (2.41)

where u =∑l−1

k=1(mk + 1)+ (i+ 1) and Din is the matrix obtained from

Dn by replacing the ith column of Dn by[Kn

(c1,y

),K(1,0)

n

(c1,y

), . . . ,K(m1,0)

n

(c1,y

),

Kn

(c2,y

),K(1,0)

n

(c2,y

), . . . ,K(mm,0)

n

(cm,y

)]T.

(2.42)

Proof. If deg(p) ≤ n, then p(x) =∑n

i=0(〈τ,pRi〉/〈τ,R2i 〉)Ri(x) so that

⟨τ,L(0,k)

n (x,y)p(x)⟩

=

n∑i=0

⟨τ,pRi

⟩⟨τ,R2

i

⟩ ⟨τ,L(0,k)

n (x,y)Ri(x)⟩

=

n∑i=0

⟨τ,pRi

⟩⟨τ,R2

i

⟩ n∑j=0

R(k)j (y)⟨τ,R2

j

⟩ ⟨τ,Rj(x)Ri(x)

=

n∑i=0

⟨τ,pRi

⟩⟨τ,R2

i

⟩ R(k)i (y) = p(k)(y).

(2.43)

Expand Ln(x,y) as Ln(x,y) =∑n

j=0 anj(y)Pj(x), where

anj(y) =

⟨σ,Ln(x,y)Pj(x)

⟩⟨σ,P2

j

⟩=

⟨τ,Ln(x,y)Pj(x)

⟩⟨σ,P2

j

⟩ −λ⟨

σ,P2j

⟩×

m∑l=1

ml∑k=0

(−1)kulk

k!

⟨δ(k)

(x−cl

),Ln(x,y)Pj(x)

=Pj(y)⟨σ,P2

j

⟩ −λ⟨

σ,P2j

⟩ m∑l=1

ml∑k=0

ulk

k!

k∑i=0

(k

i

)L(i,0)

n

(cl,y

)P

(k−i)j

(cl

)(2.44)

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80 Generalized Uvarov transforms

by (2.40). Hence

Ln(x,y) = Kn(x,y)−λ

m∑l=1

ml∑k=0

ulk

k!

k∑i=0

(k

i

)L(i,0)

n

(cl,y

)K(0,k−i)

n

(x,cl

)

= Kn(x,y)−λ

m∑l=1

ml∑i=0

ml−i∑j=0

ul,i+j

i!j!K(0,j)

n

(x,cl

)L(i,0)

n

(cl,y

),

(2.45)

and so

Dn

[Ln

(c1,y

),L(1,0)

n

(c1,y

), . . . ,L(m1,0)

n

(c1,y

),

Ln

(c2,y

), . . . ,L(mm,0)

n

(cm,y

)]T

=[Kn

(c1,y

),K(1,0)

n

(c1,y

), . . . ,K(m1,0)

n

(c1,y

),

Kn

(c2,y

), . . . ,K(mm,0)

n

(cm,y

)]T.

(2.46)

Hence, we have (2.41) from (2.45) and (2.46). �

3. Semi-classical case

Since τ is a linear spectral transform (see [29]) of σ, if σ is a Laguerre-Hahn form (see [22]) or a semi-classical form (see [24]) or a second degreeform (see [26]), then so is τ. Here, we consider the semi-classical case moreclosely.

Definition 3.1 (see Maroni [24]). A moment functional σ is said to be semi-classical if σ is quasi-definite and satisfies a Pearson type functional equa-tion (

φ(x)σ)′

−ψ(x)σ = 0 (3.1)

for some polynomials φ(x) and ψ(x) with deg(φ) ≥ 0 and deg(ψ) ≥ 1.

For a semi-classical moment functional σ, we call

s := minmax(deg(φ)−2,deg(ψ)−1

)(3.2)

the class number of σ, where the minimum is taken over all pairs (φ,ψ) ofpolynomials satisfying (3.1). We then call the MOPS {Pn(x)}∞n=0 relative to σ

a semi-classical OPS (SCOPS) of class s.From now on, we assume that σ is a semi-classical moment functional

satisfying (3.1) and set s := max(deg(φ) − 2,deg(ψ) − 1). Then τ satisfiesthe functional equation(

T(x)φ(x)τ)′

=(T

′(x)φ(x)+T(x)ψ(x)

)τ, (3.3)

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D. H. Kim and K. H. Kwon 81

where

T(x) =

m∏l=1

(x−cl

)ml+2. (3.4)

We now determine the class number of τ. By (3.3), if σ is of class s, thenτ is of class ≤ s+

∑ml=1(ml +2).

Lemma 3.2 (see [25]). The semi-classical moment functional σ satisfying(3.1) is of class s if and only if for any zero c of φ(x),

N(σ;c) :=∣∣rc

∣∣+ ∣∣⟨σ,qc(x)⟩∣∣ �= 0, (3.5)

where φ(x) = (x−c)φc(x) and φc(x)−ψ(x) = (x−c)qc(x)+rc.

Theorem 3.3. Assume that σ is of class s = max(deg(φ)− 2,deg(ψ)− 1).Then τ is of class s+

∑ml=1(ml +2) if φ(cl) �= 0, 1 ≤ l ≤ m.

Proof. Assume φ(cl) �= 0, 1 ≤ l ≤ m. Let φ(x) = T(x)φ(x) and ψ(x) =

T′(x)φ(x)+ T(x)ψ(x). For any zero c of φ(x), let φ(x) = (x− c)φc(x) and

φc(x)−ψ(x) = (x−c)qc(x)+ rc. Then either c = ct (1 ≤ t ≤ m) or φ(c) = 0.If c = ct (1 ≤ t ≤ m), then

φc(x)−ψ(x) =T(x)φ(x)

x−ct−T

′(x)φ(x)−T(x)ψ(x) =

(x−ct

)qc(x). (3.6)

Hence, rc = 0 but

⟨τ, qc(x)

⟩=

⟨σ+λ

m∑l=1

ml∑k=0

(−1)kulk

k!δ(k)

(x−cl

), qc(x)

=

⟨σ+λ

m∑l=1

ml∑k=0

(−1)kulk

k!δ(k)

(x−cl

),

T(x)φ(x)(x−ct

)2−

T′(x)φ(x)+T(x)ψ(x)

x−ct

=

⟨(φσ)

′−ψσ,

T(x)

x−ct

+

⟨λ

m∑l=1

ml∑k=0

(−1)kulk

k!δ(k)

(x−cl

),

T(x)φ(x)(x−ct

)2−

T′(x)φ(x)+T(x)ψ(x)

x−ct

= −λut,mt

(mt +1

) m∏l=1l�=t

(ct −cl

(ct

) �= 0,

(3.7)

so that N(τ,c) �= 0.

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82 Generalized Uvarov transforms

If c �= ct (1 ≤ t ≤ m), then φ(c) = 0, φc(x) = T(x)φc(x), and

φc(x)−ψ(x) = T(x)φc(x)−T′(x)φ(x)−T(x)ψ(x). (3.8)

Hence, rc = T(c)φc(c)− T(c)ψ(c) = T(c)rc. If rc �= 0, then rc �= 0 so thatN(τ;c) �= 0.

If rc = 0, then 〈σ,qc(x)〉 �= 0 and rc = 0 so that

qc(x) = T(x)qc(x)−T′(x)φc(x). (3.9)

Then ⟨τ, qc(x)

⟩=

⟨σ,qc(x)

⟩=

⟨σ,T(x)qc(x)−T

′(x)φc(x)

⟩. (3.10)

Set Q1(x), Q2(x), Q3(x), and r1, r2, r3 to be

T(x) = (x−c)Q1(x)+r1;

T′(x) = (x−c)Q2(x)+r2;

Q1(x) = (x−c)Q3(x)+r3.

(3.11)

Then Q2(x) = Q′1(x)+Q3(x) and r2 = r3 = Q1(c).

Hence,⟨τ, qc(x)

⟩=

⟨σ,Q1(x)

(φc(x)−ψ(x)

)⟩+

⟨σ,r1qc(x)

⟩−

⟨σ,Q2(x)φ(x)

⟩−

⟨σ,r2φc(x)

⟩=

⟨σ,Q3(x)φ(x)

⟩+

⟨σ,r3φc(x)

⟩−

⟨σ,Q1(x)ψ(x)

⟩+

⟨σ,r1qc(x)

⟩−

⟨σ,Q2(x)φ(x)

⟩−

⟨σ,r2φc(x)

⟩=

⟨φ(x)σ,Q3(x)

⟩+

⟨φ(x)σ,Q

′1(x)

⟩−

⟨φ(x)σ,Q2(x)

⟩+r1

⟨σ,qc(x)

⟩= r1

⟨σ,qc(x)

⟩=

m∏l=1

(c−cl

)ml+2⟨σ,qc(x)

⟩ �= 0.

(3.12)

Hence N(τ;c) �= 0. �

4. Examples

As illustrating examples, we consider the following example.

Example 4.1. Let

τ := σ+λ(u10δ(x−1)+u20δ(x+1)+u11δ′(x−1)+u21δ

′(x+1)

), (4.1)

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D. H. Kim and K. H. Kwon 83

where λ �= 0, |u10| + |u20| + |u11| + |u21| �= 0, and σ is the Jacobi momentfunctional defined by

〈σ,π〉 =

∫1

−1

(1−x)α(1+x)βπ(x)dx (α > −1,β > −1), π ∈ P. (4.2)

Then

Pn(x) = P(α,β)n (x)

=

(2n+α+β

n

)−1 n∑k=0

(n+α

n−k

)(n+β

k

)(x−1)k(x+1)n−k, n ≥ 0

(4.3)

are the Jacobi polynomials satisfying(1−x2

)y′′(x)+

[β−α−(α+β+2)x

]y′(x)+n(n+α+β+1)y(x) = 0,⟨

σ,P(α,β)n (x)2

⟩:= kn

=2α+β+2n+1n!Γ(n+α+1)Γ(n+β+1)

Γ(n+α+β+1)(2n+α+β+1)(n+α+β+1)2n

, n ≥ 0,

(4.4)

where

(a)k =

{1, if k = 0

a(a+1) · · ·(a+k−1), if k ≥ 1.(4.5)

In this case, using the differential-difference relation

(P(α,β)

n (x))(ν)

=n!

(n−ν)!P

(α+ν,β+ν)n−ν (x), ν = 0,1,2, . . . , n ≥ ν, (4.6)

the structure relation(1−x2

)P(α,β)

n (x)′ = αnP(α,β)n+1 (x)+ βnP(α,β)

n (x)+ γnP(α,β)n−1 (x), n ≥ 0,

(4.7)where

αn = −n,

βn =2(α−β)n(n+α+β+1)

(2n+α+β)(2n+2+α+β),

γn =4n(n+α)(n+β)(n+α+β)(n+α+β+1)

(2n+α+β−1)(2n+α+β)2(2n+α+β+1),

(4.8)

and the three term recurrence relation

P(α,β)n+1 (x) =

(x−βn

)P(α,β)

n (x)−γnP(α,β)n−1 (x), (4.9)

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84 Generalized Uvarov transforms

where

βn =β2 −α2

(2n+α+β)(2n+2+α+β),

γn =4n(n+α)(n+β)(n+α+β)

(2n+α+β−1)(2n+α+β)2(2n+α+β+1),

(4.10)

we can easily obtain (see [1, equations (30)–(32)]):

K(0,0)n−1 (1,1) =

(P

(α,β)n (1)

)2n(n+β)

kn−1(2n+α+β+1)γn(α+1),

K(0,0)n−1 (1,−1) = −

nP(α,β)n (−1)P

(α,β)n (1)

kn−1(2n+α+β+1)γn,

K(0,1)n−1 (1,1) =

(P

(α,β)n

)′(1)P

(α,β)n (1)(n+β)(n−1)

kn−1(2n+α+β+1)γn(α+2),

K(0,1)n−1 (1,−1) = −

(P

(α,β)n

)′(−1)P

(α,β)n (1)(n−1)

kn−1(2n+α+β+1)γn,

K(1,1)n−1 (1,1) = P(α,β)

n (1)(P(α,β)

n

)′(1)(n−1)(n+β)

×[(α+2)

(n2 +nα+nβ

)−(α+1)(α+β+2)

]2kn−1(2n+α+β+1)γn(α+1)(α+2)(α+3)

,

K(0,0)n−1 (1,1) = −

P(α,β)n (1)

(P

(α,β)n

)′(−1)(n−1)

[n2 +nα+nβ−α−β−2

]2kn−1(2n+α+β+1)γn(α+1)

,

(4.11)

where

Kn(x,y) =

n∑k=0

P(α,β)k (x)P

(α,β)k (y)

kn(4.12)

is the nth kernel polynomial of {P(α,β)n (x)}∞n=0 and K

(i,j)n (x,y)=∂i

x∂jyKn(x,y).

Using the symmetry of the Jacobi kernels, we obtain that the moment func-tional τ in (4.1) is quasi-definite if and only if

dn =

∣∣∣∣A11 A12

A21 A22

∣∣∣∣ �= 0, n ≥ 0, (4.13)

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D. H. Kim and K. H. Kwon 85

where

A11 =

(1+λu10K

(0,0)n (1,1)+λu11K

(0,1)n (1,1) λu11K

(0,0)n (1,1)

λu10K(1,0)n (1,1)+λu11K

(1,1)n (1,1) 1+λu11K

(1,0)n (1,1)

)

A12 =

(λu20K

(0,0)n (1,−1)+λu21K

(0,1)n (1,−1) λu21K

(0,0)n (1,−1)

λu20K(1,0)n (1,−1)+λu21K

(1,1)n (1,−1) λu21K

(1,0)n (1,−1)

)

A21 =

(λu10K

(0,0)n (−1,1)+λu11K

(0,1)n (−1,1) λu11K

(0,0)n (−1,1)

λu10K(1,0)n (−1,1)+λu11K

(1,1)n (−1,1) λu11K

(1,0)n (−1,1)

)

A22 =(1+λu20K

(0,0)n (−1,−1)+λu21K

(0,1)n (−1,−1) λu21K

(0,0)n (−1,−1)

λu20K(1,0)n (−1,−1)+λu21K

(1,1)n (−1,−1) 1+λu21K

(1,0)n (−1,−1)

).

(4.14)

Alvarez-Nodarse, J. Arvesu, and F. Marcellan [1] showed that for any val-ues of λ and u10, u20, u11, u21, dn �= 0 for large n so that Rn(x) exists forlarge n. Moreover, they express Rn(x) as

Rn(x) =(1+nζn+nηn

)P(α,β)

n (x)+[ζn(1−x)−ηn(1+x)+θn

](P(α,β)

n (x))′

+[χn(1+x)−ωn(1−x)

](P(α,β)

n (x))′′

,

(4.15)

where ζn, ηn, θn, χn, and ωn are constants depending on n, λ, u10, u20,

and u11, (see [1, equations (47)–(50)]). They also express Rn(x) as a gener-alized hypergeometric series 6F5 (see [1, Proposition 2]).

Example 4.2. Consider a moment functional τ given by

τ := σ+λ

N∑k=0

(−1)kuk

k!δ(k)(x), (4.16)

where λ �= 0, uk ∈ C, N ∈ {0,1,2, . . . } and σ is the Laguerre moment func-tional defined by

〈σ,p〉 =

∫∞0

xαe−xπ(x)dx (α > −1), π ∈ P. (4.17)

Then

Pn(x) = L(α)n (x) = (−1)nn!

n∑k=0

(n+α

n−k

)(−x)k

k!

= (−1)n(α+1)n 1F1(−n;α+1;x), n ≥ 0

(4.18)

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86 Generalized Uvarov transforms

are the monic Laguerre polynomials satisfying

xy′′(x)+(1+α−x)y′(x)+ny(x) = 0,⟨σ,L(α)

n (x)2⟩

= n!Γ(n+α+1), n ≥ 0.(4.19)

Hence

L(α)n (0) =

(−1)nΓ(n+α+1)

Γ(α+1)= (−1)n

(n+α

n

)n!. (4.20)

Hence, by Theorem 2.2, the moment functional τ in (4.16) is quasi-definiteif and only if dn �= 0, where dn is the determinant of the (N+1)× (N+1)

matrix Dn,

Dn :=

[δij +λ

N−j∑k=0

uj+k

j!k!K(i,k)

n (0,0)

]N

i,j=0

, n ≥ 0, (4.21)

where Kn(x,y) =∑n

k=0 L(α)k (x)L

(α)k (y)/〈σ,L

(α)k (x)2〉.

When dn �= 0 for n ≥ 0, we now claim that the MOPS {Rn(x)}∞n=0 relativeto τ can be given as

Rn(x) =

N+1∑k=0

A(n)k ∂k

xL(α)n (x), n ≥ 0 (4.22)

for suitable constants A(n)k (0 ≤ k ≤ N + 1) with A

(n)0 = 1. For any fixed

n ≥ 1, set

Rn(x) :=

N+1∑k=0

Ak∂kxL(α)

n (x), (4.23)

where {Ak}N+1k=0 are constants to be determined. Note here that if 0 ≤ n ≤ N,

then ∂kxL

(α)n (x) = 0 for n + 1 ≤ k ≤ N + 1 so that we may take Ak for

n + 1 ≤ k ≤ N + 1 to be 0. Since (L(α)n (x))′ = nL

(α+1)n−1 (x), n ≥ 1, we

have

Rn(x) =

N+1∑k=0

(n−k+1)kAkL(α+k)n−k (x), (4.24)

where L(α)n (x) = 0 for n < 0. We now show that the coefficients {Ak}N+1

k=0 can

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D. H. Kim and K. H. Kwon 87

be chosen so that

⟨τ,xmRn(x)

⟩= 0, 0 ≤ m ≤ n−1. (4.25)

If N+1 ≤ m < n, then by (4.24)

⟨τ,xmRn(x)

⟩=

∫∞0

xmRn(x)xαe−x dx+λ

N∑k=0

uk

k!(xmRn(x))(k)(0)

=

N+1∑k=0

n!

(n−k)!Ak

∫∞0

xmL(α+k)n−k (x)xαe−x dx

=

N+1∑k=0

n!

(n−k)!Ak

∫∞0

xm−kL(α+k)n−k (x)xα+ke−x dx

= 0.

(4.26)

We now assume that 0 ≤ m ≤ min(N,n−1). Then

⟨σ,xmL

(α+k)n−k (x)

⟩=

∫∞0

xmL(α+k)n−k (x)xαe−x dx

=

∫∞0

xm−kL(α+k)n−k (x)xα+ke−x dx = 0,

(4.27)

for 0 ≤ k ≤ m. For m+1 ≤ k ≤ n,

⟨σ,xmL

(α+k)n−k (x)

⟩= (−1)n−k(n−k)!

n−k∑j=0

(−1)j

j!

(n+α

n−k−j

)∫∞0

xm+α+je−x dx

= (−1)n−k(n−k)!

n−k∑j=0

(−1)j

j!

(n+α

n−k− j

)Γ(m+α+ j+1)

= (−1)n−k(n−k)!

(n+α

n−k

)Γ(m+α+1)

×2F1(−n+k,m+α+1;k+α+1;1)

= (−1)n−k(n−k)!

(n−m−1

n−k

)Γ(m+α+1)

(4.28)

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88 Generalized Uvarov transforms

by (4.18) and 2F1(−n,b;c;1) = (c−b)n/(c)n. Hence by (4.20)

⟨τ,xmRn(x)

⟩=

N+1∑k=0

(n−k+1)kAk

⟨σ,xmL

(α+k)n−k (x)

N∑l=0

ul

l!

N+1∑k=0

(n−k+1)kAk

(xmL

(α+k)n−k (x)

)(l)(0)

= n!Γ(m+α+1)

N+1∑k=m+1

(−1)n−k

(n−m−1

n−k

)Ak

+λn!

N∑l=m

ul

(l−m)!

N+1∑k=0

(−1)n−k−l+m

(n+α

n−k− l+m

)Ak,

0 ≤ m ≤ min(N,n−1),

(4.29)

where(nk

)= 0, for k < 0. Hence 〈τ,xmRn(x)〉 = 0, with 0 ≤ m ≤ n−1 if and

only if

Γ(m+α+1)

N+1∑k=m+1

(−1)n−k

(n−m−1

n−k

)Ak

N∑l=m

ul

(l−m)!

N+1∑k=0

(−1)n−k−l+m

(n+α

n−k− l+m

)Ak = 0,

0 ≤ m ≤ min(N,n−1).

(4.30)

Since (4.30) is a homogeneous system of N + 1 (resp., n) equations forN + 2 (resp., n + 1) unknowns {Ak}N+1

k=0 (resp., {Ak}nk=0) when n ≥ N + 1

(resp., n ≤ N ), there always exists a nontrivial solution {Ak}N+1k=0 . With

this choice of {Ak}N+1k=0 , Rn(x) is a nonzero polynomial of degree ≤ n and

〈τ,xmRn(x)〉 = 0 for 0 ≤ m ≤ n− 1 so that deg(Rn) = n, that is, A0 �= 0.Then A−1

0 Rn(x) = Rn(x) so that we have (4.22).Now we can express Rn(x) as a hypergeometric series (see [13]);

Rn(x) =β0β1 · · ·βN

(α+1)N+1(−1)

n(α+1)n

(A0+A1+ · · ·+AN+1

)

× N+2FN+2

(−n,β0+1,β1+1, . . . ,βN+1

α+N+2,β0,β1, . . . ,βN

∣∣∣∣x) (4.31)

for suitable constants {βj}Nj=0.

Page 21: Quasi-definiteness of generalized Uvarov transforms of moment … · 2019. 8. 1. · 70 Generalized Uvarov transforms where P n is the space of polynomials of degree ≤ n with P−1

D. H. Kim and K. H. Kwon 89

Acknowledgements

D. H. Kim was partially supported by BK-21 project and K. H. Kwon waspartially supported by KOSEF (99-2-101-001-5).

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D. H. Kim: Division of Applied Mathematics, Korea Advanced Institute of Scienceand Technology, Taejon 305-701, Korea

E-mail address: [email protected]

K. H. Kwon: Division of Applied Mathematics, Korea Advanced Institute of Scienceand Technology, Taejon 305-701, Korea

E-mail address: [email protected]

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