Classical multiple orthogonal polynomials for arbitrary number of weights and their explicit representation

TL;DR

This paper provides explicit formulas for classical multiple orthogonal polynomials with any number of weights, including recurrence coefficients and type I polynomials, advancing the theoretical understanding of these polynomial families.

Contribution

It introduces new explicit expressions for recurrence coefficients and type I polynomials for various classical multiple orthogonal polynomial families.

Findings

Explicit recurrence coefficients for all polynomial families.

New formulas for type I multiple orthogonal polynomials.

Explicit representations for the step line case.

Abstract

This paper delves into classical multiple orthogonal polynomials with an arbitrary number of weights, including Jacobi-Pi\~neiro, Laguerre of both first and second kinds, as well as multiple orthogonal Hermite polynomials. Novel explicit expressions for nearest-neighbor recurrence coefficients, as well as the step line case, are provided for all these polynomial families. Furthermore, new explicit expressions for type I multiple orthogonal polynomials are derived for Laguerre of the second kind and also for multiple Hermite polynomials.