Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation

Details Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation

2011-06-15

arxiv.org/abs/1106.2959v1

Mathematics

Classical Analysis and ODEs

14 pages

Scientific paper

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.

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