On linearly related orthogonal polynomials in several variables

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 $${\mathbb{Q}}_n={\mathbb{P}}_n+M_n{\mathbb{P}}_{n-1},n\ge 1,$$ 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.

28 pages. To appear in Numerical Algorithms

References in corpus (1)

• Orthogonal polynomials generated by a linear structure relation: Inverse problem

Cited by in corpus (2)

• Multivariate Orthogonal Polynomials and Modified Moment Functionals
• Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies