Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation

Abstract

We investigate generalizations of the Charlier polynomials on the lattice N \mathbb {N} , on the shifted lattice N + 1 − β \mathbb {N}+1-\beta , and on the bi-lattice N ∪ \mathbb {N}\;\cup ( N + 1 − β ) (\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 V _{\mathrm {V}} (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of P V _{\mathrm {V}} with special values of the parameters.