Recurrence Coefficients of a New Generalization of the Meixner Polynomials

Details Recurrence Coefficients of a New Generalization of the Meixner Polynomials Recurrence Coefficients of a New Generalization of the Meixner Polynomials

2011-04-19

arxiv.org/abs/1104.3773v2

SIGMA 7 (2011), 068, 11 pages

Mathematics

Classical Analysis and ODEs

Scientific paper

10.3842/SIGMA.2011.068

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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