SIGMA 7 (2011), 068, 11 pages      arXiv:1104.3773      https://doi.org/10.3842/SIGMA.2011.068
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Galina Filipuk ^a and Walter Van Assche ^b
a) Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, Warsaw, 02-097, Poland
^b) Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Received April 18, 2011, in final form July 07, 2011; Published online July 13, 2011

Abstract
We investigate new generalizations of the Meixner polynomials on the lattice N, on the shifted lattice N+1−β and on the bi-lattice N∪(N+1−β). 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_V. Initial conditions for different lattices can be transformed to the classical solutions of P_V with special values of the parameters. We also study one property of the Bäcklund transformation of P_V.

Key words: Painlevé equations; Bäcklund transformations; classical solutions; orthogonal polynomials; recurrence coefficients.

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