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Bilinear generating functions for orthogonal polynomials
H.T. Koelink,J. Van der Jeugt +1 more
TLDR
In this paper, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner-Pollaczek polynomial.Abstract:
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.read more
Citations
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Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an overview
TL;DR: In this article, an ordered pair of linear transformations (i.e., a Leonard pair on a field and a vector space over a field with finite positive dimension) is considered.
Proceedings ArticleDOI
Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations
TL;DR: The Tridiagonal algebra as discussed by the authors is an algebra on two generators which is defined as follows: a field is a field, and a sequence of scalars taken from a field can be represented by two symbols A and A. The corresponding Tridiagonal algebra T is the associative K-algebra with 1 generated by A. In the first part of this paper, we survey what is known about irreducible finite di-mensional T-modules.
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Leonard pairs and the Askey-Wilson relations
TL;DR: In this article, the Askey-Wilson relations were used to define the Leonard pair on a vector space over a field with finite positive dimension, where the dimension of the vector space is at least 4.
Journal ArticleDOI
Introduction to Leonard pairs
TL;DR: In this article, the authors give an elementary introduction to the theory of Leonard pairs, defined as an ordered pair of linear transformations that satisfy conditions (i), (ii) below.
Book ChapterDOI
An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials
TL;DR: In this paper, a correspondence between Leonard pairs and a class of orthogonal polynomials is discussed, which coincides with the terminating branch of the Askey scheme and consists of the q-Racah, q-Hahn, dual q-Chen, dual Hahn, Q-Krawtchouk, dual Krawchkouk, Krawczyk, Bannai/Ito, and orphan polynomial.
References
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Book
Basic Hypergeometric Series
George Gasper,Mizan Rahman +1 more
TL;DR: In this article, the Askey-Wilson q-beta integral and some associated formulas were used to generate bilinear generating functions for basic orthogonal polynomials.
Book
A guide to quantum groups
Vyjayanthi Chari,Andrew Pressley +1 more
TL;DR: In this paper, the Kac-Moody algebras and quasitriangular Hopf algesas were used to represent the universal R-matrix and the root of unity case.
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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
Roelof Koekoek,René F. Swarttouw +1 more
TL;DR: The Askey-scheme of hypergeometric orthogonal polynomials was introduced in this paper, where the q-analogues of the polynomial classes in the Askey scheme are given.
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Convolutions for orthogonal polynomials from Lie and quantum algebra representations.
H.T. Koelink,J. Van der Jeugt +1 more