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arXiv · 1104.3773

Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Galina Filipuk·Walter Van Assche
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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$_{\textup V}$.

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Galina Filipuk, Walter Van Assche. 2011-04-19. Recurrence Coefficients of a New Generalization of the Meixner Polynomials. https://doi.org/10.3842/sigma.2011.068

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