Convolutions for orthogonal polynomials from Lie and quantum algebra representations.
H.T. Koelink,J. Van der Jeugt +1 more
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In this article, the authors derived convolution identities for Al-Salam and Chihara polynomials by using the Clebsch-Gordan decomposition and the Racah coefficients.Abstract:
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.read more
Citations
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Introduction to orthogonal polynomials
TL;DR: In this paper, the authors proposed a model for parameterized t > 0: J− en = √ n(t + n− 1) en−1, J+ en = (n + 1)(t+ n) en+1 + 1, J0 en = n + 2n)en + cJ0 = X
Journal ArticleDOI
A guide to quantum groups
TL;DR: Chari and Pressley as mentioned in this paper have published a book called "Chari, Pressley, and Chari: A Conversation with Vyjayanthi Chari and Andrew Pressley".
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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 paper, the authors consider an ordered pair of linear transformations A:V→V and A*:V→ V which satisfy the following two properties: (i) there exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix represented A is diagonal.
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.