Recurrence coefficients of generalized Charlier polynomials and the fifth Painlev\'e equation

TL;DR

This paper explores generalized Charlier polynomials on various lattices and reveals that their recurrence coefficients are connected to solutions of the fifth Painlevé equation, with initial conditions linked to classical solutions.

Contribution

It establishes a novel relationship between recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation, extending understanding of their integrable structure.

Findings

Recurrence coefficients relate to solutions of the fifth Painlevé equation.

Initial conditions correspond to classical solutions of PV with special parameters.

The study covers polynomials on multiple lattice types.

Abstract

We investigate generalizations of the Charlier polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-\beta$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-\beta)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to solutions of the fifth Painlev\'e equation PV (which can be transformed to the third Painlev\'e equation). Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters.