Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation

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 PV (which can be transformed to the third Painlevé equation). Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters.

14 pages

References in corpus (2)

• Orthogonal polynomials on a bi-lattice
• Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Cited by in corpus (8)

• The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation
• Discrete semiclassical orthogonal polynomials of class one
• Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
• Classical Solutions of the Degenerate Fifth Painlevé Equation
• Ladder operators and differential equations for multiple orthogonal polynomials
• Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI
• Polynomial Sequences Associated with the Moments of Hypergeometric Weights
• Recurrence relations for the generalized Laguerre and Charlier orthogonal polynomials and discrete Painlevé equations on the $D_6^{(1)}$ Sakai surface