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Recurrence Coefficients of a New Generalization of the Meixner Polynomials

By Galina Filipuk and Walter Van Assche

Abstract

We investigate new generalizations of the Meixner polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-\beta$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-\beta)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to the solutions of the fifth Painlev\'e equation P$_{\textup V}$. Initial conditions for different lattices can be transformed to the classical solutions of P$_{\textup V}$ with special values of the parameters. We also study one property of the B\"acklund transformation of P$_{\textup V}$

Topics: Mathematics - Classical Analysis and ODEs, Mathematical Physics, 34M55, 33E17, 33C47, 42C05, 64Q30
Year: 2011
DOI identifier: 10.3842/SIGMA.2011.068
OAI identifier: oai:arXiv.org:1104.3773
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