Classical discrete multiple orthogonal polynomials: hypergeometric and integral representations

2024/09/24 by Amílcar Branquinho, Branquinho, Amílcar, Juan E. F. Díaz +7
Mathematics · #33C45 #33C47 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2409.16254

openalex publication_date 2024/09/24 · openalex created_date 2024/10/26 · openalex updated_date 2026/08/01

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.

Related

• Hahn multiple orthogonal polynomials of type I: Hypergeometrical expressions
• Integral and hypergeometric representations for multiple orthogonal polynomials
• Classical multiple orthogonal polynomials for arbitrary number of weights and their explicit representation
• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Generalizations of generating functions for higher continuous hypergeometric orthogonal polynomials in the Askey scheme