Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation

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 Painlevé equation P $_{\textup V}$ (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of P $_{\textup V}$ with special values of the parameters.