Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases

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.