Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 048, 14 pages      arXiv:1602.02724      https://doi.org/10.3842/SIGMA.2016.048
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases

Luc Vinet a and Alexei Zhedanov b
a) Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7 Canada
b) Institute for Physics and Technology, 83114 Donetsk, Ukraine

Received February 08, 2016, in final form May 07, 2016; Published online May 14, 2016; Reference [17] added May 22, 2016

Abstract
We introduce the notion of ''hypergeometric'' polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = \lambda_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis $\varphi_n(x)$: $L \varphi_n(x) = \lambda_n \varphi_n(x) + \tau_n(x) \varphi_{n-1}(x)$ with some coefficients $\lambda_n$, $\tau_n$. We find the necessary and sufficient conditions for the polynomials $P_n(x)$ to be orthogonal. For the special cases where the sets $\lambda_n$ correspond to the classical grids, we find the complete solution to these conditions and observe that it leads to the most general Askey-Wilson polynomials and their special and degenerate classes.

Key words: abstract hypergeometric operator; orthogonal polynomials; classical orthogonal polynomials.

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