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
consider a point masses perturbation τ
polynomial system {Pn (x)}∞
n=0 , we
ml
k
(k)
of σ given by τ := σ + λ m
(x − cl ), where λ,
k=0 ((−1) ulk /k!)δ
l=1
ulk , and cl are constants with ci = cj for i = j. That is, τ is a generalized
Uvarov transform of σ satisfying A(x)τ = A(x)σ, where A(x) =
m
ml +1
. We find necessary and sufficient conditions for τ to be
l=1 (x − cl )
quasi-definite. We also discuss various properties of monic orthogonal polynomial system {Rn (x)}∞
n=0 relative to τ including two examples.
1. Introduction
In the study of Padé approximation (see [5, 10, 21]) of Stieltjes type meromorphic functions
b
a
m m
l
dµ(x) k!
+
Clk k+1 ,
z−x
z − cl
(1.1)
l=1 k=0
where −∞ ≤ a < b ≤ ∞, Clk are constants, and dµ(x) is a positive Stieltjes
measure, the denominators Rn (x) of the main diagonal sequence of Padé
approximants satisfy the orthogonality
b
a
Rn (x)π(x)dµ(x) +
ml
m Clk ∂k πRn cl = 0,
l=1 k=0
c 2001 Hindawi Publishing Corporation
Copyright Journal of Applied Mathematics 1:2 (2001) 69–90
2000 Mathematics Subject Classification: 33C45
URL: http://jam.hindawi.com/volume-1/S1110757X01000225.html
π ∈ Pn−1 ,
(1.2)
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 +
ml
m (−1)k Clk δ(k) x − cl ,
(1.3)
l=1 k=0
which is a point masses perturbation of dµ(x). Orthogonality to a positive
or signed measure perturbed by one or two point masses arises naturally
also in orthogonal polynomial eigenfunctions of higher order (≥ 4) ordinary differential equations (see [14, 15, 16, 19]), which generalize Bochner’s
classification of classical orthogonal polynomials (see [6, 18]). On the other
hand, many authors have studied various aspects of orthogonal polynomials
with 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]) moment
functionals. In this work, we consider the most general such situation. That
is, we consider a moment functional τ given by
τ := σ + λ
ml
m (−1)k ulk
l=1 k=0
k!
δ(k) x − cl ,
(1.4)
where σ is a given quasi-definite moment functional, λ, ulk , and cl are
complex numbers with ul,ml = 0 and ci = cj for i = j, that is, τ is obtained
from σ by adding a distribution with finite support. We give necessary and
sufficient conditions for τ to be quasi-definite. When τ is also quasi-definite,
we discuss various properties of orthogonal polynomials {Rn (x)}∞
n=0 relative
to 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 and
P = n≥0 Pn . For any π(x) in P, let deg(π) be the degree of π(x) with the
τ (i.e., linear
convention that deg(0) = −1. For the moment functionals σ,
k
functionals on P) (see [7]), c in C, and a polynomial φ(x) = n
k=0 ak x , let
σ , π := − σ, π ;
φσ, π := σ, φπ;
π(x) − π(c)
(x − c)−1 σ, π := σ, θc π ;
;
θc π (x) :=
(2.1)
x−c
n
n
(σφ)(x) :=
aj σjk xk ;
στ, π = σ, τπ, π ∈ P.
k=0
j=k
D. H. Kim and K. H. Kwon 71
We also let
∞
σn
zn+1
F(σ)(z) :=
(2.2)
n=0
be the (formal) Stieltjes function of σ, where σn := σ, xn (n ≥ 0) are the
moments 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 a
rational (resp., linear) spectral transform of σ (resp., when C(z) = 0). Then
S(A, B, C, D)F(σ) = F(τ) if and only if
xA(x)σ = C(x)(στ) + xD(x)τ,
(2.4)
σ, A + x σθ0 A (x) + xB(x) = (στ) θ0 C (x) + τ, D + x τθ0 D (x).
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,
A(z)F(σ) + B(z)
,
F(τ) := U ck , βk · · · U c1 , β1 F(σ) =
A(z)
where A(z) =
k
i=1 (z − ci ),
B(z) =
k
i=1 βi
k
A(x)τ = A(x)σ.
j=1 (z − cj ),
j=i
(2.6)
and by (2.4)
(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
A(z)F(σ) + τθ0 A (z) − σθ0 A (z)
F(τ) =
(2.8)
A(z)
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 given
by (1.4), where σ is a quasi-definite moment functional. Let {Pn (x)}∞
n=0 be
the 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) − cn Pn−1 (x),
n ≥ 0, P−1 (x) = 0 .
(2.9)
ml +1
, τ is a generalized
Since (1.4) implies (2.7) with A(x) = m
l=1 (x − cl )
Uvarov transform of σ. Then our main concern is to find conditions under
which a generalized Uvarov transform τ, given by (1.4), of σ is also quasidefinite. In other words, we are to solve the division problem (2.7) of the
moment functionals.
Let
Kn (x, y) :=
n
Pj (x)Pj (y)
,
σ, Pj2
j=0
n≥0
(2.10)
(i,j)
i
j
be the nth kernel polynomial for {Pn (x)}∞
n=0 and Kn (x, y)=∂x ∂y Kn (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 vectors in Cn . Then
n
det In + VW t = 1 +
xj yj ,
n ≥ 1,
(2.11)
j=1
where In is the n × n identity matrix.
Theorem 2.2. The moment functional τ is quasi-definite
mif and only if
d
=
0
,
n
≥
0
,
where
d
is
the
determinant
of
(
n
n
l=1 (ml + 1)) ×
m
( l=1 (ml + 1)) matrix Dn :
m
Dn := Atl (n) t,l=1 ,
n ≥ 0,
(2.12)
where
Atl (n) = δtl δki + λ
m
l −i
j=0
ul,i+j (k,j) Kn
ct , cl
i!j!
mt
ml
.
(2.13)
k=0, i=0
If τ is quasi-definite, then the MOPS {Rn (x)}∞
n=0 relative to τ is given by
Rn (x) = Pn (x) − λ
ml m
m l −i
ul,i+j
l=1 i=0 j=0
i!j!
(0,j) Kn−1 x, cl R(i)
n cl ,
(2.14)
D. H. Kim and K. H. Kwon 73
(i)
ml
where {Rn (cl )}m
l=1,i=0 are given by
 


Pn c1
Rn c1
 R c
  P c 

 

n 1
n 1

 

..
..

 


 

.
.
 (m1 )   (m1 ) 



c1  =  Pn
c1 
Dn−1  Rn
,
 R c   P c 

 

n 2
n 2

 

..
..

 


 

.
.
(mm ) (mm ) cm
cm
Rn
Pn
Moreover,
dn 2
τ, R2n =
σ, Pn
,
dn−1
n ≥ 0 D−1 = I .
n ≥ 0 d−1 = 1 .
(2.15)
(2.16)
Proof. (⇒). Assume that τ is quasi-definite and expand Rn (x) as
Rn (x) = Pn (x) +
n−1
Cnj Pj (x),
n ≥ 1,
(2.17)
j=0
where Cnj = σ, Rn Pj /σ, Pj2 , with 0 ≤ j ≤ n − 1. Here,
ml
m (−1)k ulk (k) δ
σ, Rn Pj = τ − λ
x − cl , Rn Pj
k!
l=1 k=0
ml
m k ulk k (i) (k−i) = −λ
cl Pj
cl
R
i n
k!
l=1 k=0
so that
Rn (x) = Pn (x) − λ
n−1
j=0
= Pn (x) − λ
i=0
m ml
k P (x) ulk k (i) (k−i) j 2
cl Pj
cl
R
σ, Pj l=1 k=0 k! i=0 i n
ml
m k ulk k
l=1 k=0
= Pn (x) − λ
(2.18)
k!
i=0
i
ml m
m l −i
ul,i+j
l=1 i=0 j=0
i!j!
(0,k−i) x, cl
R(i)
n cl Kn−1
(0,j) Kn−1 x, cl R(i)
n cl .
(2.19)
Hence, we have (2.14). Set the matrices Bl and El to be




Rn cl
Pn cl
  

 Rn cl 
 P c 
 , El =  n . l  , 1 ≤ l ≤ m.
Bl = 
.




..
..




(ml ) (ml ) cl
cl
Rn
Pn
(2.20)
74
Generalized Uvarov transforms
Then,


 
B1
E1
E 
B 
 2  2
 .  = Atl (n − 1) m  .  ,
 . 
t,l=1  . 
 . 
 . 
Em
Bm
(2.21)
which gives (2.15). Now,
m
Dn = Atl (n) t,l=1
λ
= Dn−1 + 2
σ, Pn
m −i
l
ul,i+j
j=0
i!j!
(j)
Pn
(k) cl Pn ct
mt
ml
m
k=0, i=0 t,l=1

E1
 E 
1 −1
 2  m1 u1j (j) m
u1,j+1 (j) λ


P
Pn c1 , . . . ,
c
,
= Dn−1 + 1
.
n


2
.
j!
j!
σ, Pn  .  j=0
j=0

Em
m
2
u1,m1 u2j (j) Pn c1 ,
P
c2 , . . . ,
m1 !
j! n
j=0
um,mm Pn cm
mm !


B1

 B  m1
1 −1

 2  u1j (j) m
λ
u1,j+1 (j) 


= Dn−1 I +
Pn c1 ,
Pn c1 , . . . ,
2  .. 
j!
j!
σ,
P
n  .  j=0

j=0

Bm
m
2
u1,m1 u2j (j) Pn c1 ,
P
c2 , . . . ,
m1 !
j! n
j=0

um,mm 

Pn cm 
mm !

(2.22)
D. H. Kim and K. H. Kwon 75
so that
dn = dn−1
ml m
m l −i
ul,i+j (j) (i) λ
cl Rn cl
P
1+ 2
σ, Pn l=1 i=0 j=0 i!j! n
(2.23)
by Lemma 2.1. On the other hand,
τ, R2n = τ, Rn Pn
= σ, Rn Pn + λ
ml
m (−1)k ulk
l=1 k=0
δ
k!
(k)
x − cl , Rn Pn
ml
m k ulk k (j) (k−j) 2
+λ
cl
= σ, Pn
R cl Pn
j n
k!
l=1 k=0
(2.24)
j=0
ml ml
m ulk k (j) (k−j) 2
= σ, Pn
+λ
cl
R cl Pn
k! j n
l=1 j=0 k=j
so that
τ, R2n
=
2
σ, Pn
+λ
ml m
m l −j
ul,j+k
l=1 j=0 k=0
j!k!
(k) R(j)
cl .
n cl Pn
(2.25)
Hence, from (2.23) and (2.25), we have
2
σ, Pn
dn = dn−1 τ, R2n ,
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
m m
l
(−1)k ulk (k) δ
τ, Rn Pr = σ, Rn Pr + λ
x − cl , Rn Pr
k!
l=1 k=0
= σ, Rn Pr + λ
ml
m l=1 k=0
= σ, Pn Pr − λ
k ulk k (j) (k−j) cl
R cl Pr
k!
j n
j=0
ml m
m l −j
l=1 j=0 k=0
+λ
ml m
m l −j
ul,j+k
l=1 j=0 k=0
j!k!
ul,j+k (j) (0,k) Rn cl σ, Kn−1 x, cl Pr (x)
j!k!
(k) R(j)
cl
n cl Pr
ml m
m l −j
ul,j+k (j) (k) Rn cl Pr cl 1 − δnr
= σ, Pn Pr − λ
j!k!
l=1 j=0 k=0
+λ
ml m
m l −j
l=1 j=0 k=0
=
=

0,



ul,j+k (j) (k) R cl Pr cl
j!k! n
0 ≤ r ≤ n − 1,
ml m
m l −j
ul,j+k (k) (j)  σ, P2 +λ
P
cl Rn cl ,

n

j!k! n


0,
0 ≤ r ≤ n − 1,
dn 2

σ, Pn
= 0,

dn−1
(0,k)
r = n,
l=1 j=0 k=0
r = n,
(2.27)
(k)
since σ, Kn−1 (x, cl )Pr (x) = Pr (cl )(1 − δnr ). Hence,
0, if 0 ≤ m ≤ n − 1,
τ, Rn Rm = τ, Rn Pn = 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 2.2
includes the following as special cases.
• m = 1, m1 = 0: Marcellán and Maroni [23],
• m = 2, m1 = m2 = 0: Draı̈di and Maroni [9], Kwon and Park [20],
D. H. Kim and K. H. Kwon 77
• m = 1, m1 = 1: Belmehdi and Marcellán [4],
• m = 1: Kim, Kwon, and Park [12].
Some other special cases where σ is a classical moment functional were
handled in [2, 1, 3, 8, 11, 14].
From now on, we always assume that dn = 0, with n ≥ 0, so that τ is also
quasi-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) − γn Rn−1 (x), n ≥ 0,
(2.30)
where
βn = bn + ml m
m l −i
ul,i+j
λ
2
σ, Pn l=1 i=0 j=0 i!j!
(2.31)
(j) (i) (j) cl Rn+1 cl − Pn−1 cl R(i)
× Pn
n cl
γn =
dn dn−2
cn
d2n−1
(n ≥ 0),
(n ≥ 1).
(2.32)
(ii) (the quasi-orthogonality)
m
x − cl
ml +1
Rn (x) =
l=1
n+r
Cnj Pj (x),
n ≥ r,
m
Cn,n−r = 0, and
m m +1
σ, l=1 x − cl l Rn Pj
Cnj =
σ, Pj2
m m +1
τ, l=1 x − cl l Rn Pj
=
, where n − r ≤ j ≤ n + r.
σ, Pj2
where r =
(2.33)
j=n−r
l=1 (ml + 1),
(2.34)
Proof. For (i), by (2.14), we can rewrite (2.30) as
Pn+1 (x) − λ
ml m
m l −i
ul,i+j
l=1 i=0 j=0
i!j!
(i) K(0,j)
x, cl Rn+1 cl
n
ml m
m l −i
(i) ul,i+j (0,j) = x − βn Pn (x) − λ
K
x, cl Rn cl
i!j! n−1
l=1 i=0 j=0
ml m
m l −i
(i) ul,i+j (0,j) − γn Pn−1 (x) − λ
x, cl Rn−1 cl .
K
i!j! n−2
l=1 i=0 j=0
(2.35)
78
Generalized Uvarov transforms
After multiplying (2.35) by Pn (x) and applying σ, we have
−λ
ml m
m l −i
ul,i+j
i!j!
l=1 i=0 j=0
= bn − βn
(i) (j)
Pn
cl Rn+1 cl
2
σ, Pn
−λ
ml m
m l −i
ul,i+j
l=1 i=0 j=0
i!j!
(j) Pn−1 cl R(i)
n cl .
(2.36)
Hence, we have (2.31) and by (2.16)
τ, R2n
d d
= n 2 n−2 cn (n ≥ 1).
γn = (2.37)
dn−1
τ, R2n−1
n+r
m
ml +1
Rn (x) = j=0 Cnj Pj (x), where r = l=1(ml +1)
For (ii), m
l=1(x−cl )
and
m
m
+1
l
x − cl
Cnk σ, Pk2 = σ,
Rn (x)Pk (x)
=
=
l=1
m
x − cl
l=1
m
τ,
ml +1
x − cl
τ, Rn (x)Pk (x)
ml +1
(2.38)
Rn (x)Pk (x)
= 0,
if r + k < n.
l=1
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 true
interval
of the orthogonality of σ. Then
ml +1
Rn (x) has at least n−r distinct nodal zeros (i.e.,
(i) m
l=1 (x−cl )
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 σ̃ := m
l=1 (x −
be
the
MOPS
cl )ml +1 σ is also positive-definite on [ξ, η]. Let {P̃n (x)}∞
n=0
relative to σ̃. Then we may write Rn (x) = n
(x)
,
where
C̃
P̃
j=0 nj j
m
ml +1
2
x − cl
τ, Rn P̃k
C̃nk σ̃, P̃k = σ̃, Rn P̃k =
=
l=1
m
τ,
l=1
x − cl
ml +1
Rn P̃k .
(2.39)
D. H. Kim and K. H. Kwon 79
Hence, C̃nk = 0, 0 ≤ k ≤ n−r−1 so that Rn (x) = n
j=n−r C̃nj P̃j (x). Hence,
Rn (x) has at least n − r distinct nodal zeros in (ξ, η). In case ξ ≥ cl (1 ≤
m
l ≤ m), σ̃ = l=1 (x − cl )ml +1 σ is also positive-definite on [ξ, η] so that by
the same reasoning as above Rn (x) has at least n − r distinct nodal zeros
in (ξ, η).
Theorem 2.5. For any polynomial p(x) of degree at most n, we have
(0,k)
τ, Ln (x, y)p(x) = p(k) (y),
(2.40)
n
where Ln (x, y) = i=0 Ri (x)Ri (y)/τ, R2i , n ≥ 0, is the nth kernel polynomial for {Rn (x)}∞
n=0 and
Ln (x, y) = Kn (x, y) −
m ml
l −i
u m
ul,i+j (0,j) λ D n Kn
x, cl ,
dn
i!j!
l=1 i=0
(2.41)
j=0
i
where u = l−1
k=1 (mk + 1) + (i + 1) and Dn is the matrix obtained from
Dn by replacing the ith column of Dn by
(m1 ,0)
Kn c1 , y , K(1,0)
c1 , y , . . . , Kn
c1 , y ,
n
(2.42)
T
(mm ,0)
c2 , y , . . . , Kn
cm , y .
Kn c2 , y , K(1,0)
n
2
Proof. If deg(p) ≤ n, then p(x) = n
i=0 (τ, pRi /τ, Ri )Ri (x) so that
n
(0,k)
τ, pRi (0,k)
τ, Ln (x, y)Ri (x)
τ, Ln (x, y)p(x) =
2
τ, Ri
i=0
n (k)
n
τ, pRi Rj (y) τ, Rj (x)Ri (x)
=
(2.43)
2
2
τ, Ri j=0 τ, Rj
i=0
n τ, pRi (k)
Ri (y) = p(k) (y).
=
2
τ,
R
i
i=0
Expand Ln (x, y) as Ln (x, y) = n
j=0 anj (y)Pj (x), where
σ, Ln (x, y)Pj (x)
anj (y) =
σ, Pj2
τ, Ln (x, y)Pj (x)
λ
=
−
σ, Pj2
σ, Pj2
ml
m (−1)k ulk (k) δ
x − cl , Ln (x, y)Pj (x)
k!
l=1 k=0
m ml
k (k−i) ulk k (i,0) Pj (y)
λ cl , y Pj
cl
Ln
=
−
2
2
σ, Pj
σ, Pj l=1 k=0 k! i=0 i
(2.44)
×
80
Generalized Uvarov transforms
by (2.40). Hence
ml
m k ulk k (i,0) cl , y K(0,k−i)
x, cl
Ln (x, y) = Kn (x, y) − λ
L
n
i n
k!
l=1 k=0
= Kn (x, y) − λ
i=0
ml m
m l −i
ul,i+j
l=1 i=0 j=0
i!j!
K(0,j)
x, cl L(i,0)
cl , y ,
n
n
(2.45)
and so
(m1 ,0)
c1 , y , . . . , Ln
c1 , y ,
Dn Ln c1 , y , L(1,0)
n
T
(mm ,0)
cm , y
Ln c2 , y , . . . , Ln
(m1 ,0)
c1 , y , . . . , Kn
c1 , y ,
= Kn c1 , y , K(1,0)
n
T
(mm ,0)
cm , y .
Kn c2 , y , . . . , Kn
(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 LaguerreHahn form (see [22]) or a semi-classical form (see [24]) or a second degree
form (see [26]), then so is τ. Here, we consider the semi-classical case more
closely.
Definition 3.1 (see Maroni [24]). A moment functional σ is said to be semiclassical if σ is quasi-definite and satisfies a Pearson type functional equation
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 := min max deg(φ) − 2, deg(ψ) − 1
(3.2)
the class number of σ, where the minimum is taken over all pairs (φ, ψ) of
polynomials 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 satisfies
the 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
x − cl
ml +2
.
(3.4)
l=1
We now determine
mthe class number of τ. By (3.3), if σ is of class s, then
τ is of class ≤ s + l=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 + m
l=1 (ml + 2) if φ(cl ) = 0, 1 ≤ l ≤ m.
Proof. Assume φ(cl ) = 0, 1 ≤ l ≤ m. Let φ̃(x) = T (x)φ(x) and ψ̃(x) =
T (x)φ(x) + T (x)ψ(x). For any zero c of φ̃(x), let φ̃(x) = (x − c)φ̃c (x) and
φ̃c (x)− ψ̃(x) = (x−c)q̃c (x)+r̃c . Then either c = ct (1 ≤ t ≤ m) or φ(c) = 0.
If c = ct (1 ≤ t ≤ m), then
T (x)φ(x)
− T (x)φ(x) − T (x)ψ(x) = x − ct q̃c (x). (3.6)
φ̃c (x) − ψ̃(x) =
x − ct
Hence, r̃c = 0 but
ml
m (−1)k ulk (k) δ
τ, q̃c (x) = σ + λ
x − cl , q̃c (x)
k!
l=1 k=0
ml
m (−1)k ulk (k) δ
x − cl ,
= σ+λ
k!
l=1 k=0
T (x)φ(x) T (x)φ(x) + T (x)ψ(x)
2 −
x − ct
x − ct
T (x)
= (φσ) − ψσ,
(3.7)
x − ct
m m
l
(−1)k ulk (k) + λ
δ
x − cl ,
k!
l=1 k=0
T (x)φ(x) T (x)φ(x) + T (x)ψ(x)
2 −
x − ct
x − ct
m
ct − cl φ ct = 0,
= −λut,mt mt + 1
l=1
l=t
so that N(τ, c) = 0.
82
Generalized Uvarov transforms
If c = ct (1 ≤ t ≤ m), then φ(c) = 0, φ̃c (x) = T (x)φc (x), and
φ̃c (x) − ψ̃(x) = T (x)φc (x) − T (x)φ(x) − T (x)ψ(x).
(3.8)
Hence, r̃c = T (c)φc (c) − T (c)ψ(c) = T (c)rc . If rc = 0, then r̃c = 0 so that
N(τ; c) = 0.
If rc = 0, then σ, qc (x) = 0 and r̃c = 0 so that
q̃c (x) = T (x)qc (x) − T (x)φc (x).
(3.9)
Then
τ, q̃c (x) = σ, q̃c (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 ;
(3.11)
Q1 (x) = (x − c)Q3 (x) + r3 .
Then Q2 (x) = Q1 (x) + Q3 (x) and r2 = r3 = Q1 (c).
Hence,
τ, q̃c (x) = σ, Q1 (x) φc (x) − ψ(x) + σ, r1 qc (x)
− σ, Q2 (x)φ(x) − σ, r2 φc (x)
= σ, Q3 (x)φ(x) + σ, r3 φc (x)
− σ, Q1 (x)ψ(x) + σ, r1 qc (x)
− σ, Q2 (x)φ(x) − σ, r2 φc (x)
= φ(x)σ, Q3 (x) + φ(x)σ, Q1 (x)
− φ(x)σ, Q2 (x) + r1 σ, qc (x)
(3.12)
m
m +2 = r1 σ, qc (x) =
c − cl l
σ, qc (x) = 0.
l=1
Hence N(τ; c) = 0.
4. Examples
As illustrating examples, we consider the following example.
Example 4.1. Let
τ := σ + λ u10 δ(x − 1) + u20 δ(x + 1) + u11 δ (x − 1) + u21 δ (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 moment
functional defined by
1
σ, π =
(1 − x)α (1 + x)β π(x)dx (α > −1, β > −1), π ∈ P.
(4.2)
−1
Then
(α,β)
Pn (x) = Pn
(x)
−1 n 2n + α + β
n+α n+β
=
(x − 1)k (x + 1)n−k ,
n
n−k
k
n≥0
k=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,
(α,β)
σ, Pn
(x)2 := kn
=
2α+β+2n+1 n!Γ (n + α + 1)Γ (n + β + 1)
,
Γ (n + α + β + 1)(2n + α + β + 1)(n + α + β + 1)2n
where
(a)k =
n ≥ 0,
(4.4)
1,
if k = 0
a(a + 1) · · · (a + k − 1),
(4.5)
if k ≥ 1.
In this case, using the differential-difference relation
(ν)
(α,β)
Pn
(x)
=
n!
(α+ν,β+ν)
P
(x),
(n − ν)! n−ν
ν = 0, 1, 2, . . . , n ≥ ν,
the structure relation
(α,β) (α,β)
(α,β)
(α,β)
1 − x2 Pn
(x) = α̃n Pn+1 (x) + β̃n Pn
(x) + γ̃n Pn−1 (x),
(4.6)
n ≥ 0,
(4.7)
where
α̃n = −n,
β̃n =
2(α − β)n(n + α + β + 1)
,
(2n + α + β)(2n + 2 + α + β)
γ̃n =
4n(n + α)(n + β)(n + α + β)(n + α + β + 1)
,
(2n + α + β − 1)(2n + α + β)2 (2n + α + β + 1)
and the three term recurrence relation
(α,β)
(α,β)
(α,β)
Pn+1 (x) = x − βn Pn
(x) − γn Pn−1 (x),
(4.8)
(4.9)
84
Generalized Uvarov transforms
where
βn =
β2 − α2
,
(2n + α + β)(2n + 2 + α + β)
4n(n + α)(n + β)(n + α + β)
,
γn =
(2n + α + β − 1)(2n + α + β)2 (2n + α + β + 1)
(4.10)
we can easily obtain (see [1, equations (30)–(32)]):
(0,0)
Kn−1 (1, 1)
2
(α,β)
Pn
(1) n(n + β)
,
=
kn−1 (2n + α + β + 1)γn (α + 1)
(α,β)
(α,β)
(−1)Pn
(1)
nPn
,
kn−1 (2n + α + β + 1)γn
(α,β) (α,β)
Pn
(1)Pn
(1)(n + β)(n − 1)
(0,1)
,
Kn−1 (1, 1) =
kn−1 (2n + α + β + 1)γn (α + 2)
(α,β) (α,β)
Pn
(−1)Pn
(1)(n − 1)
(0,1)
Kn−1 (1, −1) = −
,
kn−1 (2n + α + β + 1)γn
(α,β) (1,1)
(α,β)
Kn−1 (1, 1) = Pn
(1) Pn
(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)
(α,β) (α,β)
Pn
(1) Pn
(−1)(n − 1) n2 + nα + nβ − α − β − 2
(0,0)
Kn−1 (1, 1) = −
,
2kn−1 (2n + α + β + 1)γn (α + 1)
(4.11)
(0,0)
Kn−1 (1, −1) = −
where
Kn (x, y) =
n
(α,β)
(α,β)
P
(x)P
(y)
k
k=0
k
kn
(α,β)
(4.12)
(i,j)
i j
is the nth kernel polynomial of {Pn (x)}∞
n=0 and Kn (x, y) = ∂x ∂y Kn (x, y).
Using the symmetry of the Jacobi kernels, we obtain that the moment functional τ in (4.1) is quasi-definite if and only if
A
dn = 11
A21
A12 = 0,
A22 n ≥ 0,
(4.13)
D. H. Kim and K. H. Kwon 85
where
A11 =
A12 =
A21 =
(0,0)
(0,1)
1 + λu10 Kn (1, 1) + λu11 Kn (1, 1)
(1,0)
λu10 Kn
(1,1)
(1, 1) + λu11 Kn
(1,1)
λu20 Kn (1, −1) + λu21 Kn (1, −1)
(0,0)
(0,1)
λu10 Kn (−1, 1) + λu11 Kn (−1, 1)
(1,0)
λu10 Kn
(1,1)
(−1, 1) + λu11 Kn
(−1, 1)
A22 =
(0,0)
(0,1)
1 + λu20 Kn (−1, −1) + λu21 Kn (−1, −1)
(1,0)
λu20 Kn
(1,1)
(−1, −1) + λu21 Kn
(1, 1)
(1,0)
(1, 1)
(0,0)
(0,1)
λu20 Kn (1, −1) + λu21 Kn (1, −1)
(1,0)
(0,0)
λu11 Kn
(−1, −1)
1 + λu11 Kn
(1, 1)
(0,0)
λu21 Kn (1, −1)
(1,0)
(1, −1)
(0,0)
(−1, 1)
(1,0)
(−1, 1)
λu21 Kn
λu11 Kn
λu11 Kn
(0,0)
λu21 Kn
(−1, −1)
(1,0)
1 + λu21 Kn
.
(−1, −1)
(4.14)
Álvarez-Nodarse, J. Arvesú, and F. Marcellán [1] showed that for any values of λ and u10 , u20 , u11 , u21 , dn = 0 for large n so that Rn (x) exists for
large n. Moreover, they express Rn (x) as
(α,β)
(α,β) Rn (x) = 1 + nζn +nηn Pn
(x) + ζn (1−x) − ηn (1 + x) + θn Pn
(x)
(α,β) + χn (1 + x) − ωn (1 − x) Pn
(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 generalized hypergeometric series 6 F5 (see [1, Proposition 2]).
Example 4.2. Consider a moment functional τ given by
τ := σ + λ
N
(−1)k uk
k=0
k!
δ(k) (x),
(4.16)
where λ = 0, uk ∈ C, N ∈ {0, 1, 2, . . . } and σ is the Laguerre moment functional defined by
∞
σ, p =
xα e−x π(x)dx (α > −1), π ∈ P.
(4.17)
0
Then
n
Pn (x) = L(α)
n (x) = (−1) n!
n
n n + α (−x)k
k=0
n−k
= (−1) (α + 1)n 1 F1 (−n; α + 1; x),
k!
n≥0
(4.18)
86
Generalized Uvarov transforms
are the monic Laguerre polynomials satisfying
xy (x) + (1 + α − x)y (x) + ny(x) = 0,
2
σ, L(α)
= n!Γ (n + α + 1), n ≥ 0.
n (x)
(4.19)
Hence
L(α)
n (0) =
n+α
(−1)n Γ (n + α + 1)
= (−1)n
n!.
n
Γ (α + 1)
(4.20)
Hence, by Theorem 2.2, the moment functional τ in (4.16) is quasi-definite
if 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
N
uj+k (i,k)
K
(0, 0)
j!k! n
,
n ≥ 0,
(4.21)
i,j=0
(α)
(α)
(α)
2
where Kn (x, y) = n
k=0 Lk (x)Lk (y)/σ, Lk (x) .
When dn = 0 for n ≥ 0, we now claim that the MOPS {Rn (x)}∞
n=0 relative
to τ can be given as
Rn (x) =
N+1
k=0
(n)
for suitable constants Ak
n ≥ 1, set
(n)
Ak ∂kx L(α)
n (x),
n≥0
(4.22)
(n)
= 1. For any fixed
(0 ≤ k ≤ N + 1) with A0
R̃n (x) :=
N+1
Ak ∂kx L(α)
n (x),
(4.23)
k=0
where {Ak }N+1
k=0 are constants to be determined. Note here that if 0 ≤ n ≤ N,
(α)
then ∂kx Ln (x) = 0 for n + 1 ≤ k ≤ N + 1 so that we may take Ak for
(α)
(α+1)
n + 1 ≤ k ≤ N + 1 to be 0. Since (Ln (x)) = nLn−1 (x), n ≥ 1, we
have
R̃n (x) =
N+1
k=0
(α)
(α+k)
(n − k + 1)k Ak Ln−k (x),
(4.24)
where Ln (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, xm R̃n (x) = 0,
0 ≤ m ≤ n − 1.
(4.25)
If N + 1 ≤ m < n, then by (4.24)
τ, x R̃n (x) =
m
=
∞
0
xm R̃n (x)xα e−x dx + λ
N+1
k=0
=
N+1
k=0
N
uk
k=0
n!
Ak
(n − k)!
n!
Ak
(n − k)!
∞
0
∞
0
k!
(xm R̃n (x))(k) (0)
(α+k)
xm Ln−k (x)xα e−x dx
(4.26)
(α+k)
xm−k Ln−k (x)xα+k e−x dx
= 0.
We now assume that 0 ≤ m ≤ min(N, n − 1). Then
(α+k)
σ, xm Ln−k (x) =
=
∞
0
∞
0
(α+k)
xm Ln−k (x)xα e−x dx
(4.27)
(α+k)
xm−k Ln−k (x)xα+k e−x dx
= 0,
for 0 ≤ k ≤ m. For m + 1 ≤ k ≤ n,
(α+k)
σ, xm Ln−k (x)
n−k
= (−1)
(n−k)!
n−k
j=0
= (−1)n−k (n − k)!
n−k
j=0
∞
(−1)j n+α
xm+α+j e−x dx
j!
n−k−j 0
n+α
(−1)j
Γ (m + α + j + 1)
j!
n−k−j
n+α
Γ (m + α + 1)
= (−1)n−k (n − k)!
n−k
× 2 F1 (−n + k, m + α + 1; k + α + 1; 1)
n−m−1
= (−1)n−k (n − k)!
Γ (m + α + 1)
n−k
(4.28)
88
Generalized Uvarov transforms
by (4.18) and 2 F1 (−n, b; c; 1) = (c − b)n /(c)n . Hence by (4.20)
N+1
(α+k)
τ, xm R̃n (x) =
(n − k + 1)k Ak σ, xm Ln−k (x)
k=0
N
N+1
(l)
ul (α+k)
(n − k + 1)k Ak xm Ln−k (x)
(0)
+λ
l!
l=0
k=0
N+1
n−m−1
(−1)n−k
Ak
= n!Γ (m + α + 1)
n−k
k=m+1
+ λn!
N
l=m
N+1
n+α
ul
(−1)n−k−l+m
Ak ,
n−k−l+m
(l − m)!
k=0
0 ≤ m ≤ min(N, n − 1),
(4.29)
where
only if
n
k
= 0, for k < 0. Hence τ, xm R̃n (x) = 0, with 0 ≤ m ≤ n−1 if and
N+1
Γ (m + α + 1)
n−k
(−1)
k=m+1
+λ
N
l=m
n−m−1
Ak
n−k
N+1
n+α
ul
n−k−l+m
(−1)
Ak = 0,
n−k−l+m
(l − m)!
(4.30)
k=0
0 ≤ m ≤ min(N, n − 1).
Since (4.30) is a homogeneous system of N + 1 (resp., n) equations for
n
N + 2 (resp., n + 1) unknowns {Ak }N+1
k=0 (resp., {Ak }k=0 ) when n ≥ N + 1
(resp., n ≤ N ), there always exists a nontrivial solution {Ak }N+1
k=0 . With
,
(x)
is
a
nonzero
polynomial
of
degree
≤ n and
R̃
this choice of {Ak }N+1
n
k=0
τ, xm R̃n (x) = 0 for 0 ≤ m ≤ n − 1 so that deg(R̃n ) = n, that is, A0 = 0.
Then A−1
0 R̃n (x) = Rn (x) so that we have (4.22).
Now we can express Rn (x) as a hypergeometric series (see [13]);
Rn (x) =
β0 β1 · · · βN
n
(−1) (α + 1)n A0 +A1 + · · · + AN+1
(α + 1)N+1
−n, β0 +1, β1 +1, . . . , βN +1
x
× N+2 FN+2
α + N + 2, β0 , β1 , . . . , βN for suitable constants {βj }N
j=0 .
(4.31)
D. H. Kim and K. H. Kwon 89
Acknowledgements
D. H. Kim was partially supported by BK-21 project and K. H. Kwon was
partially supported by KOSEF (99-2-101-001-5).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
R. Álvarez-Nodarse, J. Arvesú, and F. Marcellán, A generalization of the JacobiKoornwinder polynomials, preprint.
R. Álvarez-Nodarse and F. Marcellán, A generalization of the class of Laguerre
polynomials: asymptotic properties and zeros, Appl. Anal. 62 (1996), no. 34, 349–366. Zbl 866.33005.
J. Arvesú, R. Álvarez-Nodarse, F. Marcellán, and K. H. Kwon, Some extension of
the Bessel-type orthogonal polynomials, Integral Transform. Spec. Funct. 7
(1998), no. 3-4, 191–214. MR 2001b:33013. Zbl 936.33003.
S. Belmehdi and F. Marcellán, Orthogonal polynomials associated with some
modifications of a linear functional, Appl. Anal. 46 (1992), no. 1-2, 1–24.
MR 94f:33012. Zbl 807.39013.
C. Bernardi and Y. Maday, Some spectral approximations of one-dimensional
fourth-order problems, Progress in Approximation Theory, Academic Press,
Massachusetts, 1991, pp. 43–116. MR 92j:65176.
S. Bochner, Über Sturm-Liouvillesche polynomsysteme, Math. Z. 29 (1929),
730–736.
T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its
Applications, vol. 13, Gordon and Breach Science Publishers, New York, 1978.
MR 58#1979. Zbl 389.33008.
, Orthogonal polynomials and measures with end point masses, Rocky
Mountain J. Math. 15 (1985), no. 3, 705–719. MR 87a:42037. Zbl 586.33007.
N. Draı̈di and P. Maroni, Sur l’adjonction de deux masses de Dirac à une forme
régulière quelconque [On the adjointness of two Dirac masses to an arbitrary regular form], Orthogonal Polynomials and their Applications (Spanish)
(Vigo, 1988), Esc. Téc. Super. Ing. Ind. Vigo, Vigo, 1989, pp. 83–90 (French).
MR 92m:42024. Zbl 764.42015.
A. A. Gonchar, On convergence of Pade approximants for some classes of
meromorphic functions, Math. USSR, Sb. 26 (1975), 555–575, [translated
from Mat. Sb. 97 (1975) 607–629]. Zbl 341.30029.
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.
D. H. Kim, K. H. Kwon, and S. B. Park, Delta perturbation of a moment functional, Appl. Anal. 74 (2000), no. 3-4, 463–477. MR 2001c:33018.
R. Koekoek, Generalizations of Laguerre polynomials, J. Math. Anal. Appl. 153
(1990), no. 2, 576–590. MR 92d:33016.
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.
A. M. Krall, Orthogonal polynomials satisfying fourth order differential equations, 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 differential equation, Pennsylvania State College Studies, 1940 (1940), no. 6,
24. MR 2,98a. Zbl 060.19210.
K. H. Kwon and J. H. Lee, Division problem of moment functionals, submitted.
K. H. Kwon and L. L. Littlejohn, Classification of classical orthogonal polynomials, J. Korean Math. Soc. 34 (1997), no. 4, 973–1008. MR 99k:33028.
Zbl 898.33002.
K. H. Kwon, L. L. Littlejohn, and G. J. Yoon, Orthogonal polynomial solutions
to spectral type differential equations; Magnus’ Conjecture, submitted.
K. H. Kwon and S. B. Park, Two-point masses perturbation of regular moment functionals, Indag. Math. (N.S.) 8 (1997), no. 1, 79–93. MR 99d:42044.
Zbl 877.42009.
G. L. Lopes, Convergence of Pade approximants of Stieltjes type meromorphic
functions and comparative asymptotics for orthogonal polynomials, Math.
USSR, Sb. 64 (1989), 207–227, [translated from Mat. Sb. 136 (1988) 206–
226]. Zbl 669.30026.
A. P. Magnus, Associated Askey-W ilson polynomials as Laguerre-Hahn orthogonal 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.
F. Marcellán and P. Maroni, Sur l’adjonction d’une masse de Dirac à une forme
régulière et semi-classique [On the assignment of a Dirac-mass for a regular and semi-classical form], Ann. Mat. Pura Appl. (4) 162 (1992), 1–22
(French). MR 94e:33014. Zbl 771.33008.
P. Maroni, Prolégomènes à l’étude des polynômes orthogonaux semi-classiques
[Preliminary remarks for the study of semi-classical orthogonal polynomials], Ann. Mat. Pura Appl. (4) 149 (1987), 165–184 (French). MR 89c:33016.
Zbl 636.33009.
, Variations around classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math. 48 (1993), no. 1-2, 133–155. MR 94k:33013.
Zbl 790.33006.
, An introduction to second degree forms, Adv. Comput. Math. 3 (1995),
no. 1-2, 59–88. MR 96c:42053. Zbl 837.42009.
P. G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213,
1–185. MR 80k:42025. Zbl 405.33009.
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.
A. Zhedanov, Rational spectral transformations and orthogonal polynomials, J. Comput. Appl. Math. 85 (1997), no. 1, 67–86. MR 98h:42026.
Zbl 918.42016.
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
D. H. Kim: Division of Applied Mathematics, Korea Advanced Institute of Science
and Technology, Taejon 305-701, Korea
E-mail address: dhkim@jacobi.kaist.ac.kr
K. H. Kwon: Division of Applied Mathematics, Korea Advanced Institute of Science
and Technology, Taejon 305-701, Korea
E-mail address: khkwon@jacobi.kaist.ac.kr