Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases

Abstract

We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = λ_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis $φ_n(x)$: $L φ_n(x) = λ_n φ_n(x) + τ_n(x) φ_{n-1}(x)$ with some coefficients $λ_n$, $τ_n$. We find the necessary and sufficient conditions for the polynomials $P_n(x)$ to be orthogonal. For the special cases where the sets $λ_n$ 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.