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Quadratic transformations for a function that generalizes 2F1 and the Askey-Wilson polynomials

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Abstract

In previous papers we introduced and studied a ‘relativistic’ hypergeometric function R(a +, a −, c; v, \(\hat{V}\)) that satisfies four hyperbolic difference equations of Askey-Wilson type. Specializing the family of couplings c∊\(\mathbb{C}^4\) to suitable two-dimensional subfamilies, we obtain doubling identities that may be viewed as generalized quadratic transformations. Specifically, they give rise to a quadratic transformation for 2 F 1 in the ‘nonrelativistic’ limit, and they yield quadratic transformations for the Askey-Wilson polynomials when the variables v or \(\hat{V}\) are suitably discretized. For the general coupling case, we also study the bearing of several previous results on the Askey-Wilson polynomials.

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Authors and Affiliations

  1. Centre for Mathematics and Computer Science, P.O. Box 94079, 1090 GB, Amsterdam, The Netherlands

    S. N. M. Ruijsenaars

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  1. S. N. M. Ruijsenaars
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Correspondence to S. N. M. Ruijsenaars.

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Dedicated to Richard Askey on the occasion of his 70th birthday.

2000 Mathematics Subject Classification Primary—33D45, 39A70

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Ruijsenaars, S.N.M. Quadratic transformations for a function that generalizes 2F1 and the Askey-Wilson polynomials. Ramanujan J 13, 339–364 (2007). https://doi.org/10.1007/s11139-006-0257-x

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  • DOI: https://doi.org/10.1007/s11139-006-0257-x

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