Recurrence Relations for Meixner Multiple Orthogonal Polynomials on Interlacing Lattices

A. I. Aptekarev, A. V. Dyachenko, V. G. Lysov

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia

Abstract: We consider the product of two classical Meixner weight functions whose arguments are shifted with respect to each other. Its restriction to the corresponding integer lattices generates a system of two discrete weights. Under certain conditions on the parameters, this system is perfect and has positive weights. Nevertheless, it is neither an Angelesco system nor a Nikishin system. The corresponding orthogonal polynomials are known to satisfy a four-term recurrence relation for the step-line indices. In this paper, we find explicit expressions for the coefficients of these relations. Passing to limits then yields the coefficients of four-term recurrence relations for a few other families.

Keywords: discrete orthogonal polynomials, semi-classical weight, hypergeometric weight, multiple orthogonal polynomials, Painlevé equations.

Received: March 5, 2025
Revised: June 4, 2025
Accepted: July 10, 2025

DOI: 10.4213/tm4475

 English version: Proceedings of the Steklov Institute of Mathematics, 2025, 330, 1–21

Bibliographic databases:

•