Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases

TL;DR

This paper introduces hypergeometric polynomials relative to Newtonian bases, characterizes their orthogonality conditions, and connects them to classical Askey-Wilson polynomials and their variants.

Contribution

It defines hypergeometric polynomials with respect to Newtonian bases and derives conditions for their orthogonality, linking them to classical polynomial families.

Findings

Characterization of hypergeometric polynomials in Newtonian bases.

Necessary and sufficient conditions for orthogonality.

Connection to Askey-Wilson and related polynomials.

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.