# Symmetric abstract hypergeometric polynomials

## TL;DR

This paper classifies orthogonal polynomials arising from a specific operator acting on monomials, revealing their algebraic structures as extensions of the Askey-Wilson algebra and exploring their bispectral properties.

## Contribution

It provides a complete classification of orthogonal polynomials associated with a particular operator and derives the algebraic framework explaining their bispectrality.

## Key findings

- Classification of orthogonal polynomials for the operator $L$
- Derivation of algebraic structures as extensions of Askey-Wilson algebra
- Identification of bispectrality in the polynomial family

## Abstract

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= \lambda_n x^n + \nu_n x^{n-2}$ for $\lambda_n$ and $\nu_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = \lambda_n P_n(x)$. A classification of all the cases for which the polynomials $P_n(x)$ are orthogonal is provided. A general derivation of the algebras explaining the bispectrality of the polynomials is given. The resulting algebras prove to be central extensions of the Askey-Wilson algebra and its degenerate cases.