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
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.
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
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)
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)
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)
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),
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)
⟩
+λ
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
)
+λ
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],
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)
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)
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)
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)
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
)φ
(ct
) �= 0,
(3.7)
so that N(τ,c) �= 0.
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)
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)
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)
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)
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
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)
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)
⟩
+λ
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
+λ
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.
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).
References
[1] R. Alvarez-Nodarse, J. Arvesu, and F. Marcellan, A generalization of the Jacobi-Koornwinder polynomials, preprint.
[2] R. Alvarez-Nodarse and F. Marcellan, A generalization of the class of Laguerrepolynomials: asymptotic properties and zeros, Appl. Anal. 62 (1996), no. 3-4, 349–366. Zbl 866.33005.
[3] J. Arvesu, R. Alvarez-Nodarse, F. Marcellan, and K. H. Kwon, Some extension ofthe Bessel-type orthogonal polynomials, Integral Transform. Spec. Funct. 7(1998), no. 3-4, 191–214. MR 2001b:33013. Zbl 936.33003.
[4] S. Belmehdi and F. Marcellan, Orthogonal polynomials associated with somemodifications of a linear functional, Appl. Anal. 46 (1992), no. 1-2, 1–24.MR 94f:33012. Zbl 807.39013.
[5] C. Bernardi and Y. Maday, Some spectral approximations of one-dimensionalfourth-order problems, Progress in Approximation Theory, Academic Press,Massachusetts, 1991, pp. 43–116. MR 92j:65176.
[6] S. Bochner, Uber Sturm-Liouvillesche polynomsysteme, Math. Z. 29 (1929),730–736.
[7] T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and itsApplications, vol. 13, Gordon and Breach Science Publishers, New York, 1978.MR 58#1979. Zbl 389.33008.
[8] , Orthogonal polynomials and measures with end point masses, RockyMountain J. Math. 15 (1985), no. 3, 705–719. MR 87a:42037. Zbl 586.33007.
[9] N. Draıdi and P. Maroni, Sur l’adjonction de deux masses de Dirac a une formereguliere quelconque [On the adjointness of two Dirac masses to an arbi-trary regular form], Orthogonal Polynomials and their Applications (Spanish)(Vigo, 1988), Esc. Tec. Super. Ing. Ind. Vigo, Vigo, 1989, pp. 83–90 (French).MR 92m:42024. Zbl 764.42015.
[10] A. A. Gonchar, On convergence of Pade approximants for some classes ofmeromorphic functions, Math. USSR, Sb. 26 (1975), 555–575, [translatedfrom Mat. Sb. 97 (1975) 607–629]. Zbl 341.30029.
[11] E. Hendriksen, A Bessel type orthogonal polynomial system, Nederl.Akad. Wetensch. Indag. Math. 46 (1984), no. 4, 407–414. MR 86c:33016.Zbl 564.33006.
[12] D. H. Kim, K. H. Kwon, and S. B. Park, Delta perturbation of a moment func-tional, Appl. Anal. 74 (2000), no. 3-4, 463–477. MR 2001c:33018.
[13] R. Koekoek, Generalizations of Laguerre polynomials, J. Math. Anal. Appl. 153(1990), no. 2, 576–590. MR 92d:33016.
[14] T. H. Koornwinder, Orthogonal polynomials with weight function (1−x)α(1+
x)β +Mδ(x+1)+Nδ(x−1), Canad. Math. Bull. 27 (1984), no. 2, 205–214.MR 85i:33011. Zbl 532.33010.
[15] A. M. Krall, Orthogonal polynomials satisfying fourth order differential equa-tions, Proc. Roy. Soc. Edinburgh Sect. A 87 (1980/81), no. 3-4, 271–288.MR 82d:33021. Zbl 453.33006.
90 Generalized Uvarov transforms
[16] H. L. Krall, On orthogonal polynomials satisfying a certain fourth order dif-ferential equation, Pennsylvania State College Studies, 1940 (1940), no. 6,
24. MR 2,98a. Zbl 060.19210.[17] K. H. Kwon and J. H. Lee, Division problem of moment functionals, submitted.[18] K. H. Kwon and L. L. Littlejohn, Classification of classical orthogonal poly-
nomials, J. Korean Math. Soc. 34 (1997), no. 4, 973–1008. MR 99k:33028.Zbl 898.33002.
[19] K. H. Kwon, L. L. Littlejohn, and G. J. Yoon, Orthogonal polynomial solutionsto spectral type differential equations; Magnus’ Conjecture, submitted.
[20] K. H. Kwon and S. B. Park, Two-point masses perturbation of regular mo-ment functionals, Indag. Math. (N.S.) 8 (1997), no. 1, 79–93. MR 99d:42044.Zbl 877.42009.
[21] G. L. Lopes, Convergence of Pade approximants of Stieltjes type meromorphicfunctions and comparative asymptotics for orthogonal polynomials, Math.USSR, Sb. 64 (1989), 207–227, [translated from Mat. Sb. 136 (1988) 206–226]. Zbl 669.30026.
[22] A. P. Magnus, Associated Askey-Wilson polynomials as Laguerre-Hahn orthog-onal polynomials, Orthogonal Polynomials and their Applications (Segovia,1986), Lecture Notes in Math., vol. 1329, Springer, Berlin, 1988, pp. 261–278.MR 90d:33008. Zbl 645.33015.
[23] F. Marcellan and P. Maroni, Sur l’adjonction d’une masse de Dirac a une formereguliere et semi-classique [On the assignment of a Dirac-mass for a reg-ular and semi-classical form], Ann. Mat. Pura Appl. (4) 162 (1992), 1–22(French). MR 94e:33014. Zbl 771.33008.
[24] P. Maroni, Prolegomenes a l’etude des polynomes orthogonaux semi-classiques[Preliminary remarks for the study of semi-classical orthogonal polynomi-als], Ann. Mat. Pura Appl. (4) 149 (1987), 165–184 (French). MR 89c:33016.Zbl 636.33009.
[25] , Variations around classical orthogonal polynomials. Connected prob-lems, J. Comput. Appl. Math. 48 (1993), no. 1-2, 133–155. MR 94k:33013.Zbl 790.33006.
[26] , An introduction to second degree forms, Adv. Comput. Math. 3 (1995),no. 1-2, 59–88. MR 96c:42053. Zbl 837.42009.
[27] P. G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213,
1–185. MR 80k:42025. Zbl 405.33009.[28] V. B. Uvarov, The connection between systems of polynomials orthogonal
with respect to different distribution functions, U.S.S.R. Comput. Math.and Math. Phys. 9 (1969), 25–36. Zbl 231.42013.
[29] A. Zhedanov, Rational spectral transformations and orthogonal polyno-mials, J. Comput. Appl. Math. 85 (1997), no. 1, 67–86. MR 98h:42026.Zbl 918.42016.
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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