Abstract "hypergeometric" orthogonal polynomials

TL;DR

This paper classifies all orthogonal polynomial solutions to a specific hypergeometric-type operator, revealing that only known families like Jacobi, Laguerre, Bessel, and some q-analogues satisfy these conditions.

Contribution

It provides a complete classification of orthogonal polynomials that are eigenfunctions of a two-diagonal linear operator of hypergeometric type.

Findings

Identifies Jacobi, Laguerre, Bessel, and little -1 Jacobi polynomials as the only solutions.

Shows uniqueness of polynomial eigensolutions under nondegenerate conditions.

Includes special and degenerate cases like q-analogues.

Abstract

We find all polynomials solutions $P_n(x)$ of the abstract "hypergeometric" equation $L P_n(x) = \lambda_n P_n(x)$, where $L$ is a linear operator sending any polynomial of degree $n$ to a polynomial of the same degree with the property that $L$ is two-diagonal in the monomial basis, i.e. $L x^n = \lambda_n x^n + \mu_n x^{n-1}$ with arbitrary nonzero coefficients $\lambda_n, \mu_n$ . Under obvious nondegenerate conditions, the polynomial eigensolutions $L P_n(x) = \lambda_n P_n(x)$ are unique. The main result of the paper is a classification of all {\it orthogonal} polynomials $P_n(x)$ of such type, i.e. $P_n(x)$ are assumed to be orthogonal with respect to a nondegenerate linear functional $\sigma$. We show that the only solutions are: Jacobi, Laguerre (correspondingly little $q$-Jacobi and little $q$-Laguerre and other special and degenerate cases), Bessel and little -1 Jacobi polynomials.