Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Abstract

We investigate new generalizations of the Meixner 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 the solutions of the fifth Painlevé equation P${}_{\text{V}}$. Initial conditions for different lattices can be transformed to the classical solutions of P${}_{\text{V}}$ with special values of the parameters. We also study one property of the Bäcklund transformation of P${}_{\text{V}}$.

References in corpus (2)

• The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation
• Orthogonal polynomials on a bi-lattice

Cited by in corpus (8)

• The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation
• The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation
• Discrete semiclassical orthogonal polynomials of class one
• Recurrence coefficients of generalized Charlier polynomials and the fifth Painlevé equation
• Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations
• Orthogonal and multiple orthogonal polynomials, random matrices, and Painlevé equations
• Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI
• The generalized Krawtchouk polynomials and the fifth Painlevé equation