
SIGMA 7 (2011), 068, 11 pages arXiv:1104.3773
http://dx.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, 02097, Poland
^{b)} Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B box 2400, BE3001 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 bilattice N∪(N+1−β). We show that the coefficients of the
threeterm 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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