A new class of orthogonal polynomials

Abstract

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations of the form $P_0(x)=1$, $P_1(x)=x$ and $x P_n(x)=a_n P_{n+1}(x)+c_n P_{n-1}(x)\;(n\in\mathbb{N})$, where $a_n$ and $c_n$ are positive and sum up to $1$. $(P_n(x))_{n\in\mathbb{N}_0}$ is said to satisfy nonnegative linearization of products if the product of any two polynomials $P_m(x)$, $P_n(x)$ is a convex combination of $P_{|m-n|}(x),\ldots,P_{m+n}(x)$. This property gives rise to a hypergroup structure and a sophisticated harmonic analysis. We are interested in examples such that both the original sequence $(P_n(x))_{n\in\mathbb{N}_0}$ and the sequence $(\widetilde{P_n}(x))_{n\in\mathbb{N}_0}$ which corresponds to switched roles of $(a_n)_{n\in\mathbb{N}}$ and $(c_n)_{n\in\mathbb{N}}$ satisfy nonnegative linearization of products. Such considerations were recently started by Lasser and Obermaier and can be motivated from a harmonic analytic, combinatorial or probabilistic point of view. However, Lasser and Obermaier left open the question whether examples besides the trivial example of the Chebyshev polynomials of the first kind $(T_n(x))_{n\in\mathbb{N}_0}$ (with $a_n\equiv c_n\equiv1/2$) actually exist. We provide a sufficient criterion and explicitly construct such nontrivial examples. Moreover, we provide characterizations of $(T_n(x))_{n\in\mathbb{N}_0}$ by additionally involving properties of the duals and Haar measures. Our criterion also enables us to solve open problems concerning the Haar measure of polynomial hypergroups stated by Kahler and Szwarc.