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Wilson Function Transforms Related to Racah Coefficients

Wilson Function Transforms Related to Racah Coefficients The irreducible $*$ -representations of the Lie algebra $${\mathfrak {\rm u}}(1,1)$$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch–Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for $${\mathcal U}_q({\mathfrak{\rm u}}(1,1))$$ , which turn out to be Askey–Wilson functions and Askey–Wilson polynomials. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Wilson Function Transforms Related to Racah Coefficients

Acta Applicandae Mathematicae , Volume 91 (2) – Jun 28, 2006

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References (60)

Publisher
Springer Journals
Copyright
Copyright © 2006 by Springer Science + Business Media B.V.
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-006-9024-7
Publisher site
See Article on Publisher Site

Abstract

The irreducible $*$ -representations of the Lie algebra $${\mathfrak {\rm u}}(1,1)$$ consist of discrete series representations, principal unitary series and complementary series. We calculate Racah coefficients for tensor product representations that consist of at least two discrete series representations. We use the explicit expressions for the Clebsch–Gordan coefficients as hypergeometric functions to find explicit expressions for the Racah coefficients. The Racah coefficients are Wilson polynomials and Wilson functions. This leads to natural interpretations of the Wilson function transforms. As an application several sum and integral identities are obtained involving Wilson polynomials and Wilson functions. We also compute Racah coefficients for $${\mathcal U}_q({\mathfrak{\rm u}}(1,1))$$ , which turn out to be Askey–Wilson functions and Askey–Wilson polynomials.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Jun 28, 2006

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