Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation

Filipuk, Galina; Van Assche, Walter

doi:10.1090/S0002-9939-2012-11468-6

Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation
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by Galina Filipuk and Walter Van Assche PDF

Proc. Amer. Math. Soc. 141 (2013), 551-562 Request permission

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

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