Symmetric abstract hypergeometric polynomials

Abstract

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L{x}^n=\hat{I}{»}_n{x}^n+\hat{I}{½}_n{x}^{n-2}$ for $\hat{I}{»}_n$ and $\hat{I}{½}_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L{P}_n(x)=\hat{I}{»}_n{P}_n(x)$. 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.