Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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 PV. Initial conditions for different lattices can be transformed to the classical solutions of PV with special values of the parameters. We also study one property of the Bäcklund transformation of PV.

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

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