Elementary Askey-Wilson Functions
Abstract
We present a determinantal evaluation formula for the very-well-poised 8Φ7 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 qk 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.
Related
- Askey--Wilson polynomials with ASEP parameters
- The Askey-Wilson Function is the Laplace Transform of a Positive Measure
- Bernstein-type bases on q-quadratic lattices and Askey--Wilson\slash q-Racah connection coefficients
- Fractional elementary symmetric functions and Hessian estimates in the plane
- On Integer-Division Bases for the Kalmár Elementary Functions
BibTeX
@misc{diejen2026elementary,
title = {{Elementary Askey-Wilson Functions}},
author = {J. F. van Diejen and A. N. Kirillov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37390},
eprint = {2609.37390},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37390},
}
RIS
TY - UNPB
TI - Elementary Askey-Wilson Functions
AU - J. F. van Diejen
AU - A. N. Kirillov
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37390
UR - https://arxiv.org/abs/2609.37390
AB - 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.
ER -