Ismail M., Koelink E. (editors) Theory and Applications of Special Functions

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454

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

same consideration

in the proof of Lemma 2.3 shows that the condi-

tions (1.4) of Theorem

1.1

are satisfied.

An example of the explicit expansion in the Askey-Wilson polynomials

is the generating relation found by Ismail and Wilson (Ismail and Wilson,

1982):

(ar1

br1

CT,

dr; q)m

(41 cd; q)n

(x;

c1 d)

n=O

(ab, cd, reie, re-ie; q)m

(2.43)

reie, re-ie

2v2

(

ar,

where Irl

we have used (A.3.5) of (Suslov, 2003b) in the right

side. By Theorem

the corresponding ellipse of convergence is

with

Irl;

the first poles in the right occur on the ellipse.

Some Expansions in Classical Orthogonal

Polynomials

Consider also several expansions in Chebyshev and Jacobi polynomi-

als.

3.1

The Chebyshev polynomials

The polynomials Tn (x) and Un (x) are usually defined as

sin

Tn (COS

cos

no,

Un (COS

sin 8

(34

They have several generating relations including

(x)

n=O

-2rx+r2'

where

Irl

Expansions in q-Polynomials

455

where R

2rx

r2,

Irl

and

(x)

eTx

cosh

(r&Ci)

n=O

(3.6)

see, for example, (Andrews et

al.,

1999), (Rainville, 1960).

From the definition of the Chebyshev polynomials one can establish

the following lower and upper bounds on the ellipse

with

given

by (1.2)-(1.3),

when

1,2,3,.

. .

and

when

0,1,2,

. .

Thus, by Theorem 1.1 the ellipse of convergence of

the series in (3.2)-(3.5) is

with

Irl

the series in (3.6)-(3.7)

represent entire functions.

3.2

The Jacobi polynomials

Expansions of analytic functions in series of Jacobi polynomials are

discussed in (Askey, 1975), (Boas and Buck, 1964), (Erd6lyi et al., 1953)

and (Szeg6, 1975). More details on the ellipse of convergence, asymp-

totics and inequalities can be found in (Antonov and Kholshevnikov,

1979), (Carlson, 1974b), (Carlson, 1974a) and references therein.

As an example, Jacobi's generating relation,

2"+P

P?@)

(x)

(3.10)

n=O ~(1-r+R)"(l+r+~)~'

where R

2x7-

r2)'I2, holds in the ellipse

with

Irl

Some q-Taylor's Expansions

There is certain interest nowadays in expansions of polynomials and

entire functions in the so-called q-Taylor series (Ismail and Stanton,

2002), (Ismail and Stanton, 2003a), and (Kac and Cheung, 2002); see

456

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

also Exercise 3.8 of (Suslov, 2003b). We only consider here two exam-

ples related to the basic exponential function on a q-quadratic grid. The

Taylor expansion of

(x;

with respect to

where by the definition

and

We assume that the empty products when

0 here are equal to

1. Formula (4.1) provides also an expansion of the basic exponential

function with respect to the set of polynomials

{cpn

(x; q))F==O. This ba-

sis has been used for the basic sine and cosine functions (Bustoz and

Suslov, 1998) and for the basic exponential functions on a q-quadratic

grid (Suslov, 2001)-(Suslov, 2002) and, recently, in a more general set-

ting of the q-Taylor expansions (Ismail and Stanton, 2003a)-(Ismail and

Stanton, 2003b).

One can easily establish the following upper bounds

Expansions

q-Polynomials

and

where by the definition 1x1

X(E)

(qg

qbg)/2. These inequalities

show that the polynomials

(9,

(x; q))r=o are uniformly bounded with

respect to

for any finite value of x. Thus, by Remark 1.2 the series

(4.1) is an entire function in the complex x-plane when q1/21al

The

coefficients of this series coincide with those found by the q-Taylor series

with respect to cpn(x; q). Indeed, in this case a formal q-Taylor formula

has the form (Ismail and Stanton, 2003a)-(Ismail and Stanton, 2003b)

where

and the operator Dq

S/6x is the standard first order Askey-Wilson

divided difference operator

with x(z)

(qZ

q-') /2

cos8, qz

eie;

see (Gasper and

1990) and (Suslov, 2003b) for more details. Here we took

(4.8)

Rahman,

also into

account that the basic exponential function

(x)

lq(x;

satisfies the

difference equation

which is a q-version of

on a q-quadratic grid.

In the recent papers (Ismail and Stanton, 2002)-(Ismail and Stanton,

2003a) Ismail and Stanton pointed out the following representations for

458

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

the q-exponential function

where

la1

and by the definition

Two independent proofs of (4.11)-(4.12) are presented in (Suslov, 2003b);

see Sections 2.3 and 3.4.3.

The uniform upper bound is

and, therefore, the series in (4.12) is an entire function in the complex

plane when

la1

The corresponding formal q-Taylor formula (Ismail

and Stanton, 2002)-(Ismail and Stanton, 2003a)

where

gives again the right coefficients in the expansions (4.11).

Acknowledgments

The author thanks Dick Askey, Joaquin Bustoz, George Gasper, and

Dennis Stanton for valuable discussions; he also expresses his apprecia-

tion to the referees of this paper for their important comments.

Expansions in q-Polynomials

459

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Barry M. McCoy.

STRONG NONNEGATIVE LINEARIZATION

OF ORTHOGONAL POLYNOMIALS

Ryszard Szwarc

Institute of Mathematics

Wroclaw University

pl. Grunwaldzki

2/4

50-384 Wroclaw

POLAND

and

Institute

Mathematics

Polish Academy of Science

u1. Sniadeckich

00-950 Warszawa

POLAND

Abstract

stronger notion of nonnegative linearization of orthogonal polynomi-

als is introduced. It requires that also the associated polynomials of

any order have nonnegative linearization property. This turns out to be

equivalent to a maximal principle of

discrete boundary value problem

associated with orthogonal polynomials through the three term recur-

rence relation. The property is stable for certain perturbations of the

recurrence relation. Criteria for the strong nonnegative linearization are

derived. The range of parameters for the Jacobi polynomials satisfying

this new property is determined.

Keywords:

Orthogonal polynomials, recurrence relation, nonnegative linearization,

discrete boundary value problem.

Introduction

One of the main problems in the theory of orthogonal polynomials is

to determine whether the expansion of the product of two orthogonal

*This work was partially supported by KBN (Poland) under grant

P03A 034 20 and by

European Commission via TMR network "Harmonic Analysis and Related Problems," RTN2-

2001-00315.

2005

Springer Science+Business Media, Inc.

462

THEORY AND APPLICATIONS

SPECIAL FUNCTIONS

polynomials in terms of these polynomials has nonnegative coefficients.

We want to decide which orthogonal systems {~~)r=~ have the property

with nonnegative coefficients c(n, m, k) for every n, m and

Numerous classical orthogonal polynomials as well as their q-analogues

satisfy nonnegative linearization property (Gasper, 1970a; Gasper, 1970b;

Gasper, 1983), (Gasper and Rahman, 1990), (Ramis, 1992), (Rogers,

1894), (Szwarc, 1992b; Szwarc, 1995). There are many criteria for non-

negative linearization given in terms of the coefficients of the recurrence

relation the orthogonal polynomials satisfy (Askey, 1970), (Mlotkowski

and Szwarc, 2001), (Szwarc, 1992a; Szwarc, 199213; Szwarc, 2003), that

can be applied to general orthogonal polynomials systems. These crite-

ria are based on the connection between the linearization property and

a certain discrete boundary value problem of hyperbolic type.

In this paper we are going to show that many polynomials systems

satisfy even a stronger version of nonnegative linearization. Namely let

{P,)~?~

be an orthogonal polynomial system. Let {pn )r=o denote the

associated polynomials of order

We say that the polynomials

{P~},",~

satisfy the strong nonnegative linearization property if

with nonnegative coefficients c(n, m, k) and cl(n, m, k) for any n, m, k

and

The interesting feature of this property is the fact that it is equivalent

to a maximum principle of the associated boundary value problem (see

Theorem 2). Also this property is invariant for certain transformations of

the recurrence relation (see Proposition 2), unlike the usual nonnegative

linearization property.

In the last part of this work we are going to show that the Jacobi

polynomials have the strong linearization property if and only if either

a=P>-1/2ora>P>-landa+P>O.

Strong nonnegative linearization

Let pn denote a sequence of orthogonal polynomials, relative to a

measure

satisfying the recurrence relation

Strong nonnegative linearization of orthogonal polynomials

463

where

yn,

an+l

and

/3n

We use the convention that

and

111

p-1

For any nonnegative integer

let

denote the sequence

of polynomials satisfying

111

[I1

For

the polynomial

is of degree

The polynomials

are called the

associated polynomial of order

These polynomials are

orthogonal, as well. Let

denote any orthogonality measure associated

111

with.

{pn In=,+i

For

consider the polynomials

pn(x)pm(x)

and

pI1 (x)p;(x).

We can express these products in terms of

pk(x)

to obtain the following.

The polynomial

pn(x)pm(x)

has degree

while

p!] (x)p$(x)

has

degree

Hence the expansions have finite ranges and by

the recurrence relation we obtain expansions of the form

Definition

2.1.

The system of orthogonal polynomials pn satisfies

the

strong nonnegative linearization property

(SNLP)

The form of recurrence relation used in

(2.1)

and

(2.2)

is suitable for

applications. For technical reasons we will work with the renormalized