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 ($L{P}_n(x)={\lambda }_n{P}_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis ${\phi }_n(x)$: $L{\phi }_n(x)={\lambda }_n{\phi }_n(x)+{\tau }_n(x){\phi }_{n-1}(x)$ 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.

References in corpus (1)

• Abstract "hypergeometric" orthogonal polynomials

Cited by in corpus (4)

• Charting the $q$-Askey scheme
• Meta Algebras and Biorthogonal Rational Functions: The Hahn Case
• Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$
• Symmetric abstract hypergeometric polynomials