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

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.