Geodesic
Recurrence Coefficients of a New Generalization of the Meixner Polynomials
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 Painlevé 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 Bäcklund transformation of P$_{\textup V}$.
Keywords: Painlevé equations; Bäcklund transformations; classical solutions; orthogonal polynomials; recurrence coefficients. 
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     author = {Galina Filipuk and Walter Van Assche},
     title = {Recurrence {Coefficients} of a {New} {Generalization} of the {Meixner} {Polynomials}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a67/}
}
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Galina Filipuk; Walter Van Assche. Recurrence Coefficients of a New Generalization of the Meixner Polynomials. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a67/

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