Structure relations of classical orthogonal polynomials of the quadratic and $q$-quadratic variable

Maurice Kenfack Nangho, Kerstin Jordaan

Published 2018-01-31Version 1

We characterize polynomials that satisfy a first structure relation of the form \begin{equation}\label{1} \pi(x)\mathbb{D}_{x}^2P_{n}(x)=\sum_{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\quad a_{n,n-2}\neq 0,\quad n=2,3,\dots \end{equation} where $\pi(x)$ is a polynomial of degree at most $4$ and $\mathbb{D}_{x}$ is the divided-difference operator \[\nonumber \mathbb{D}_{x}\,f(x(t))={f(x(t+{1\over 2}))-f(x(t-{1\over 2}))\over x(t+{1\over 2})-x(t-{1\over 2})}.\] We prove an equivalence between the existence of a structure relation of type (1) satisfied by a sequence of monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives $\{\mathbb{D}_{x}^2P_n\}_{n= 2}^{\infty}$ and a generalized Sturm-Liouville type equation and show that the only monic orthogonal polynomials that satisfy (1) are Wilson polynomials, Continuous Dual-Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tends to $\infty$. This work extends our previous work (cf. [17]) concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying (1).