Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations

Abstract

We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations.

21 pages, some revisions made

References in corpus (8)

• Painleve IV and degenerate Gaussian Unitary Ensembles
• The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation
• Orthogonal polynomials with discontinuous weights
• Discrete semiclassical orthogonal polynomials of class one
• Recurrence Coefficients of a New Generalization of the Meixner Polynomials
• Correlations of RMT Characteristic Polynomials and Integrability: Hermitean Matrices
• Classical transcendental solutions of the Painlevé equations and their degeneration
• The generalized Krawtchouk polynomials and the fifth Painlevé equation

Cited by in corpus (10)

• The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation
• A hidden analytic structure of the Rabi model
• A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions
• Orthogonal and multiple orthogonal polynomials, random matrices, and Painlevé equations
• Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI
• Polynomial Sequences Associated with the Moments of Hypergeometric Weights
• The Padé interpolation method applied to $q$-Painlevé equations
• The Padé interpolation method applied to $q$-Painlevé equations II (differential grid version)
• A q-generalization of the Toda equations for the q-Laguerre/Hermite orthogonal polynomials
• Determinantal approach to multiple orthogonal polynomials, and the corresponding integrable equations