Towards a \(q\)-analogue of the Kibble-Slepian formula in 3 dimensions. (English) Zbl 1236.33035
Author’s 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 3-dimensional Normal distribution. We show that this is not the case. Neverthless, 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 Askey-Wilson density disclosing its new, interesting properties. In particular we are able to obtain a generalization of the 2-dimensional Poisson-Mehler formula. We also pose several open questions.
Reviewer: Chrysoula G. Kokologiannaki (Patras)
MSC:
| 33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
Kibble-Slepian formula; Askey-Wilson density; orthogonal polynomials; \(q\)-Hermite; Al-Salam-Chihara polynomials; Poisson-Mehler kernel; positive kernelsReferences:
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