Classical discrete multiple orthogonal polynomials: hypergeometric and integral representations

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.