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Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SIGMA_2016_12_a47,
     author = {Luc Vinet and Alexei Zhedanov},
     title = {Hypergeometric {Orthogonal} {Polynomials} with respect to {Newtonian} {Bases}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a47/}
}
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Luc Vinet; Alexei Zhedanov. Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a47/

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