The generalized Krawtchouk polynomials and the fifth Painlevé equation
arXiv:1204.5070
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.
15 pages, 4 figures, references added with more details and explanations, to appear in Journal of Difference Equations and Applications