Abstract "hypergeometric" orthogonal polynomials

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.

20 pages

Cited by in corpus (3)

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