Orthogonal polynomials on a bi-lattice

TL;DR

This paper explores orthogonal polynomials on a bi-lattice formed by combining two shifted lattices, revealing their recurrence coefficients satisfy a nonlinear system related to discrete Painlevé equations.

Contribution

It introduces a new class of orthogonal polynomials on a bi-lattice and connects their recurrence relations to discrete Painlevé equations, extending previous polynomial theories.

Findings

Recurrence coefficients satisfy a nonlinear recurrence system.

Connection established between orthogonal polynomials and discrete Painlevé equations.

Generalization of Charlier and Meixner polynomials to bi-lattice setting.

Abstract

We investigate generalizations of the Charlier and the Meixner polynomials on the lattice N and on the shifted lattice N+1-\beta. We combine both lattices to obtain the bi-lattice N \cup (N+1-\beta) and show that the orthogonal polynomials on this bi-lattice have recurrence coefficients which satisfy a non-linear system of recurrence equations, which we can identify as a limiting case of an (asymmetric) discrete Painlev\'e equation.