Discrete orthogonality of the polynomial sequences in the q-Askey scheme

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arXiv:2210.13731v1 [math.CA] 25 Oct 2022

Discrete orthogonality of the polynomial sequences in the

q-Askey scheme

Luis Verde-Star

Department of Mathematics, Universidad Aut´onoma Metropolitana, Iztapalapa, Mexico City, Mexico

Abstract

We obtain weight functions associated with q-linear and q-quadratic lattices that yield

discrete orthogonality with respect to a quasi-deﬁnite moment functional for the Askey-

Wilson polynomials and all the polynomial sequences in the q-Askey scheme, with the

exception of the continuous q-Hermite polynomials.

AMS classiﬁcation: 33C45, 33D45.

Keywords: Basic hypergeometric orthogonal polynomials, discrete orthogonality, gen-

eralized moments, q-linear and q-quadratic lattices, q-Askey scheme.

1. Introduction

The families of basic hypergeometric orthogonal polynomial sequences included in

the q-Askey scheme [8] have been studied for a long time. Some of those families have

a well-known orthogonality determined by a weight function w(xk) deﬁned on a ﬁnite

or inﬁnite set of points xkthat belong to a q-linear or q-quadratic lattice. In such cases

the orthogonality of a polynomial sequence {un(t) : n∈N}is given by

k∈M

un(xk)um(xk)w(xk) = Knδn,m, n, m ∈M, (1)

where M={0,1,2,...,N}, for some positive integer N, or M=N, and Knis a nonzero

constant for n∈M.

The polynomial families that satisfy certain types of diﬀerential or diﬀerence equa-

tions and have a discrete orthogonality with positive weights w(xk) have been character-

ized and studied in great detail. The main references for discrete orthogonal polynomials

are the books [11] and [8]. See also [1], [4], [5], and [12].

In this paper, we present a construction that produces for each polynomial family

in the q-Askey scheme a weight function w(xk) that determines a discrete orthogonality

with respect to a quasi-deﬁnite moment functional. The weights w(xk) are the values

at t= 1 of a sequence of basic hypergeometric functions fk(t) that are of type 3Φ2, or

Email address: verde@xanum.uam.mx (Luis Verde-Star)

Preprint submitted to Elsevier October 26, 2022

type 2Φ1. The functions fk(t) depend on the parameters that determine the coeﬃcients

in the three-term recurrence relation of the corresponding polynomial family. The basic

hypergeometric functions fk(t) are convergent at t= 1 and we show how they are

obtained from f0(t).

For some families in the q-Askey scheme we obtain two discrete orthogonality weights

associated with diﬀerent sequences of nodes xk. In this paper we do not deal with the

problem of characterizing the cases for which the weights are positive.

We obtain our results using the linear algebraic approach of our previous papers [14],

[15], [16], and [17]. In [17] we presented a uniﬁed construction of all the hypergeomet-

ric and basic hypergeometric orthogonal polynomial sequences that uses three linearly

recurrent sequences of numbers that satisfy certain diﬀerence equation of order three.

The initial terms of such sequences determine the sequences of orthogonal polynomi-

als and provide us with a uniform parametrization of all the hypergeometric and basic

hypergeometric sequences. In the present paper we use several results from [17].

In Section 2 we present some preliminary material related with the construction of

the hypergeometric q-orthogonal polynomials and their uniform parametrization from

[17]. In Section 3 we present the construction of the weight functions obtained by solving

an inﬁnite system of linear equations and use some properties of basic hypergeometric

functions. In Section 4 we ﬁnd general formulas for the weight functions for the cases

when some of the parameters are equal to zero. In Section 5 we present explicit expres-

sions for the weight functions for some families of polynomials in the q-Askey scheme.

2. The class Hqof basic hypergeometric orthogonal polynomial sequences

In this section we present some properties of the class Hqof the basic hypergeometric

orthogonal polynomial sequences that were obtained in [17].

Consider the homogeneous diﬀerence equation

sk+3 =z(sk+2 −sk+1) + sk, k >0,(2)

where z= 1+q+q−1. The roots of the characteristic polynomial of this equation are 1, q,

and q−1. In the present paper we consider only the case with distinct roots. Therefore

the general solution of (2) is a linear combination of the sequences 1, qkand q−k. Let

the sequences xk,hk, and dkbe solutions of (2). Then we can write these sequences as

follows

xk=b0+b1qk+b2q−k, hk=a0+a1qk+a2q−k, dk=s0+s1qk+s2q−k.(3)

The sequence xkdetermines the Newtonian basis {vn(t) : n>0}of the complex

vector space of polynomials in the complex variable t, deﬁned by

vn(t) = (t−x0)(t−x1)···(t−xn−1), n >1,(4)

and v0(t) = 1.We deﬁne the sequence gkby

gk=xk−1(hk−h0) + dk, k >1,(5)

and g0= 0. Therefore we must have d0= 0 and hence s0=−s1−s2. In addition,

we suppose that hk6=hjwhenever k6=j, and that gk6= 0 for k>1. We can take

a0= 0 because a0does not appear in any of the formulas that we will obtain in our

development.

Let Dbe the linear operator on the space of polynomials deﬁned by

Dvk=hkvk+gkvk−1, k >0.(6)

Since g0= 0 we see that Dtnis equal to hntnplus a polynomial of degree less than n.

For n>0 let unbe a monic polynomial of degree nwhich is an eigenfunction of Dwith

eigenvalue hn. That is

Duk=hkuk, k >0.(7)

The operator Dis a generalized diﬀerence operator which, in concrete examples,

becomes a second order diﬀerence operator on a linear or quadratic q-lattice. In [17] we

showed that

un(t) =

k=0

cn,kvk(t), n >0,(8)

where the coeﬃcients cn,k are given by

cn,k =

n−1

j=k

gj+1

hn−hj

,06k6n−1,(9)

and cn,n = 1 for n>0. This expression for un(t) was also obtained by Vinet and

Zhedanov in [18] using a diﬀerent approach. The idea of representing orthogonal poly-

nomials in terms of a Newtonian basis was introduced by Geronimus in [7].

The matrix of coeﬃcients C= [cn,k] is an inﬁnite lower triangular matrix where the

coeﬃcients of unappear in the n-th row of C. Since cn,n = 1 for n>0, the inﬁnite

matrix Cis invertible. Let C−1= [ˆcn,k] and deﬁne the polynomials

wn,k(t) =

n−1

j=k

(t−hj),06k6n. (10)

Using basic properties of divided diﬀerences we obtain

ˆcn,k =Qn

j=k+1 gj

w′

n+1,k(hk)=

j=k+1

hk−hj

,06k6n−1,(11)

and ˆcn,n = 1 for n>0.

The entries in the 0-th column of C−1are given by

ˆck,0=

j=1

h0−hj

, k >1,(12)

and ˆc0,0= 1. We denote them by mk= ˆck,0for k>0.They satisfy m0= 1 and

k=0

cn,k mk= 0, n >1.(13)

Note that the sequence mnsatisﬁes a recurrence relation of order one. We will see

that the numbers mkare the generalized moments, with respect to the Newtonian basis

{vk(t)}, of a moment functional for which the polynomial sequence un(t) is orthogonal.

In [17] we also proved that the polynomial sequence un(t) satisﬁes a three-term

recurrence relation of the form

un+1(t) = (t−βn)un(t)−αnun−1(t), n >1,(14)

where the coeﬃcients are given by and

αn=gn

hn−1−hngn−1

hn−2−hn

−gn

hn−1−hn

+gn+1

hn−1−hn+1

+xn−xn−1,(15)

and

βn=xn+gn+1

hn−hn+1

−gn

hn−1−hn

.(16)

Writing αnand βnin terms of the parameters introduced in (3) we obtain

αn=(x−1)(a1x−qa2)p1(x)p2(x)

(a1x2−a2)(a1x2−qa2)2(a1x2−q2a2), n >1,(17)

where x=qn,

p1(x) = (a1x−a2)(b1x2+b0qx +b2q2) + qx(s1x−s2),

and

p2(x) = −a2

1x3

q2a2

p1qa2

a1z,

and

βn=qb0(x2−1)(a2

1x2−a2

2) + d x(x−1)(a1x−a2)−(qs1−s2)(a1−qa2)x2

(a1qx2−a2)(a1x2−qa2), n >0,

(18)

where

d= (q+ 1)(qa1b2+a2b1)−q(s1+s2+ (a1+a2)b0).

Let us note that αnand βnare rational functions of qnand are determined by the

parameters a1, a2, b0, b1, b2, s1, s2. Note also that at least one of a1and a2must be

diﬀerent from zero.

It is easy to verify that αnis invariant under the substitution x→qa2/(a1x) and βn

is invariant under the substitution x→a2/(a1x).

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摘要：

arXiv:2210.13731v1[math.CA]25Oct2022Discreteorthogonalityofthepolynomialsequencesintheq-AskeyschemeLuisVerde-StarDepartmentofMathematics,UniversidadAut´onomaMetropolitana,Iztapalapa,MexicoCity,MexicoAbstractWeobtainweightfunctionsassociatedwithq-linearandq-quadraticlatticesthatyielddiscreteorthogona...

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Discrete orthogonality of the polynomial sequences in the q-Askey scheme

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