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arXiv · 2609.37390

Elementary Askey-Wilson Functions

J. F. van Diejen·A. N. Kirillov
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Abstract

We present a determinantal evaluation formula for the very-well-poised ${}_8Φ_7$ basic hypergeometric Askey-Wilson function in terms of elementary functions. The formula in question is valid for discrete values of the four permutation-symmetric Askey-Wilson parameters; specifically, these consist of all quadruples of parameter values that are given, up to a sign, by powers $q^k$ with $k$ a positive integer or half-integer, in such a way that each of the four types (positive/negative sign, integer/half-integer power) occurs exactly once. The proof hinges on a product formula for an associated Cauchy-type determinant, which is of independent interest and is established here by elementary means using Krattenthaler's `identification of factors' method for determinant evaluations.

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J. F. van Diejen, A. N. Kirillov. 2026-09-29. Elementary Askey-Wilson Functions. https://arxiv.org/abs/2609.37390

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