Towards a q-analogue of the Kibble--Slepian formula in 3 dimensions

TL;DR

This paper explores a q-analogue of the Kibble-Slepian formula in three dimensions, revealing its limitations and potential applications in summing kernels related to Askey-Wilson polynomials and generalizing the Poisson-Mehler formula.

Contribution

It introduces a q-analogue of the 3D Kibble-Slepian formula, analyzes its properties, and demonstrates its use in summing kernels and deriving new properties of Askey-Wilson polynomials.

Findings

The q-analogue does not always produce non-negative values.

The formula can sum certain kernels built from Al-Salam-Chihara polynomials.

A new generalization of the 2D Poisson-Mehler formula is obtained.

Abstract

We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q-Hermite ones. If such a replacement would lead to non-negativity for all allowed values of parameters and for all values of variables ranging over certain Cartesian product of compact intervals then we would deal with a generalization of the 3 dimensional Normal distribution. We show that this is not the case. We indicate some values of the parameters and some compact set in R^{3} of positive measure, such that the values of the extension of KS formula are on this set negative. Nevertheless we indicate other applications of so generalized KS formula. Namely we use it to sum certain kernels built of the Al-Salam-Chihara polynomials for the cases that were not considered by other authors. One of such kernels sums up to the Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2 dimensional Poisson-Mehler formula. As a corollary we indicate some new interesting properties of the Askey-Wilson polynomials with complex parameters. We also pose several open questions.