Classical orthogonal polynomials in two variables: a matrix approach

Abstract

Classical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated to a two-variable moment functional satisfying a matrix analogue of the Pearson differential equation. Furthermore, we characterize classical orthogonal polynomials in two variables as the polynomial solutions of a matrix second order partial differential equation.

New steps on Sobolev orthogonality in two variables

“…In fact, we will use the gradient operator ∇, and the divergence operator div, defined as usual. The extension of these operators for matrices is introduced in [18,19]. Let A, B 0 , B 1 ∈ M h×k (P ) be polynomial matrices.…”

On differential properties for bivariate orthogonal polynomials

“…These operators can be extended for higher order derivatives [1], and for matrices [5,6]. In fact, if we denote…”

“…The case h = 1 became the usual structure relation, (see Proposition 1.1), and it characterizes classical moment functionals (see [5,6]). …”

“…In the case h = 1, in [5,6], the authors proved the reciprocal: if {P n } n≥0 is a WOPS associated with u, such that the sequence {∇P t n } n≥1 satisfy the orthogonality condition (23), then u is classical (Proposition 1.2).…”

“…Recently, the authors (see [5,6]) extended the concept of classical orthogonal polynomials in two variables to a wider framework, which, of course, includes the Krall and Sheffer definition and tensor products of classical orthogonal polynomials in one variable. Our approach to classical orthogonal polynomials in two variables starts from a quasi-definite moment functional u satisfying a matrix Pearson-type differential equation…”