Classical discrete multiple orthogonal polynomials: hypergeometric and integral representations

TL;DR

This paper provides explicit hypergeometric and integral representations for classical discrete multiple orthogonal polynomials, expanding the understanding of their structure and relationships within the Askey scheme.

Contribution

It introduces explicit hypergeometric and integral formulas for multiple orthogonal polynomials in the Hahn class, including new recursion coefficients and limit relations.

Findings

Explicit hypergeometric representations for type I multiple Hahn polynomials

Integral representations for Hahn class polynomials

Recursion coefficients derived for Hahn multiple orthogonal polynomials

Abstract

This work explores classical discrete multiple orthogonal polynomials, including Hahn, Meixner of the first and second kinds, Kravchuk, and Charlier polynomials, with an arbitrary number of weights. Explicit expressions for the recursion coefficients of Hahn multiple orthogonal polynomials are derived. By leveraging the multiple Askey scheme and the recently discovered explicit hypergeometric representation of type I multiple Hahn polynomials, corresponding explicit hypergeometric representations are provided for the type I polynomials and recursion coefficients of all the aforementioned descendants within the Askey scheme. Additionally, integral representations for these families within the Hahn class in the Askey scheme are presented. The multiple Askey scheme is further completed by providing the corresponding limits for the weights, polynomials, and recurrence coefficients.