Proceedings of the American Mathematical Society vol:141 issue:2 pages:551-562
PII S 0002-9939(2012)11468-6
We investigate generalizations of the Charlier polynomials on the lattice N, on the shifted lattice N+1-\beta and on the bi-lattice N \cup (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 (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of Painlevé V with special values of the parameters.