The generalized Krawtchouk polynomials and the fifth Painlevé equation

Abstract

We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth Painlevé equation when viewed as functions of one of the parameters in the weight.

arXiv comment15 pages, 4 figures, references added with more details and explanations, to appear in Journal of Difference Equations and Applications

References in corpus (3)

• Application of the $\tau$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE
• Orthogonal polynomials on a bi-lattice
• Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Cited by in corpus (4)

• 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
• Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI