On linearly related orthogonal polynomials in several variables

TL;DR

This paper investigates conditions under which two polynomial systems in several variables, related by a linear structure, are orthogonal, providing characterizations and examples relevant to multivariate orthogonal polynomials.

Contribution

It characterizes when two related polynomial systems are both orthogonal and when one orthogonal system implies the orthogonality of the other, in the multivariate setting.

Findings

Characterization of linear structure relations for orthogonal polynomial systems

Conditions for one polynomial system's orthogonality to imply the other's

Examples illustrating the theoretical results

Abstract

Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$ are constant matrices of proper size and $\mathbb{Q}_0 = \mathbb{P}_0$. The aim of our work is twofold. First, if both polynomial systems are orthogonal, characterize when that linear structure relation exists in terms of their moment functionals. Second, if one of the two polynomial systems is orthogonal, study when the other one is also orthogonal. Finally, some illustrative examples are presented.