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This article is cited in 4 scientific papers (total in 4 papers)
Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases
Luc Vineta, Alexei Zhedanovb 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
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.
Keywords:
abstract hypergeometric operator; orthogonal polynomials; classical orthogonal polynomials.
Received: February 8, 2016; in final form May 7, 2016; Published online May 14, 2016
Citation:
Luc Vinet, Alexei Zhedanov, “Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases”, SIGMA, 12 (2016), 048, 14 pp.
Linking options:
https://www.mathnet.ru/eng/sigma1130 https://www.mathnet.ru/eng/sigma/v12/p48
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