A new class of orthogonal polynomials

TL;DR

This paper introduces a new class of orthogonal polynomials arising from specific recurrence relations, explores their harmonic analysis properties, and constructs nontrivial examples satisfying nonnegative linearization of products, extending known results.

Contribution

The paper provides a sufficient criterion for nontrivial examples of polynomial sequences with nonnegative linearization, constructs explicit examples, and characterizes Chebyshev polynomials within this framework.

Findings

Constructed explicit nontrivial polynomial examples

Provided criteria for nonnegative linearization of products

Solved open problems on Haar measures of polynomial hypergroups

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.