Symmetric abstract hypergeometric polynomials

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.

References in corpus (7)

• A "missing" family of classical orthogonal polynomials
• The Universal Askey-Wilson Algebra
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Bispectrality of the Complementary Bannai-Ito Polynomials
• The Symmetrical $H_q$-Semiclassical Orthogonal Polynomials of Class One
• Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases
• Abstract "hypergeometric" orthogonal polynomials