We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

representations, parameterized by t > 0: J− en = √ n(t + n− 1) en−1, J+ en = √ (n + 1)(t + n) en+1, J0 en = (t + 2n)en Want J− + J+ + cJ0 = X

https://www.math.tamu.edu/~manshel/papers/Orthogonal-poly.pdf

Introduction to orthogonal polynomials

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below 
(i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal 
(ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal 
We call such a pair a Leonard pair on $V$ We give an overview of the theory of Leonard pairs

Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview

We define an algebra on two generators which we call the Tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let K denote a field, and let β, γ, γ � , , � denote a sequence of scalars taken from K. The corresponding Tridiagonal algebra T is the associative K-algebra with 1 generated by two symbols A, Asubject to the relations (A, A 2 A � − βAAA + AA 2 − γ(AA � + AA) − A � ) = 0, (A � , A �2 A − βAAA � + AA �2 − γ � (AA + AA � ) − � A) = 0, In the first part of this paper, we survey what is known about irreducible finite di- mensional T-modules. We focus on how these modules are related to the Leonard pairs recently introduced by the present author, and the more general Tridiagonal pairs recently introduced by Ito, Tanabe, and the present author. In the second part of the paper, we construct an infinite dimensional irreducible T-module based on the Askey-Wilson polynomials. This module is on the vector space K(x) consisting of all polynomials in an indeterminant x that have coefficients inK. Denoting by A the linear transformation on K(x) which is multiplication by x, and denoting by Aan

Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations

Theinterpretation of the Meixner--Pollaczek, Meixner, and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra ${\frak{su}}(1,1)$ and the Clebsch--Gordan decomposition lead to generalizations of the convolution identities for these polynomials. Using the Racah coefficients, convolution identities for continuous Hahn, Hahn, and Jacobi polynomials are obtained. From the quantized universal enveloping algebra for ${\frak{su}}(1,1)$, convolution identities for the Al-Salam and Chihara polynomials and the Askey--Wilson polynomials are derived by using the Clebsch--Gordan and Racah coefficients. For the quantized universal enveloping algebra for ${\frak{su}}(2)$, q-Racah polynomials are interpreted as Clebsch--Gordan coefficients, and the linearization coefficients for a two-parameter family of Askey--Wilson polynomials are derived.

Convolutions for orthogonal polynomials from Lie and quantum algebra representations.

Convolutions for orthogonal polynomials from Lie and quantum algebra representations

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials.

Bilinear Generating Functions for Orthogonal Polynomials

Laurent polynomials related to the Hahn-Extonq-Bessel function, which areq-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurentq-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials, the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurentq-Lommel polynomials and related functions is used to obtain estimates for the latter.

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H.T. Koelink

Papers

Convolutions for orthogonal polynomials from Lie and quantum algebra representations.

Convolutions for orthogonal polynomials from Lie and quantum algebra representations

Bilinear Generating Functions for Orthogonal Polynomials

Orthogonal polynomials and Laurent polynomials related to the Hahn-Exton q-Bessel function

Bilinear generating functions for orthogonal polynomials