On linearly related orthogonal polynomials in several variables

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.