Quadratic Decomposition of Bivariate Orthogonal Polynomials

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Backlund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.

Automated summary

We discuss the relationship between bivariate orthogonal polynomial systems associated with symmetric weight functions and those associated with specific Christoffel modifications of the quadratic decomposition of the original weight. We examine the construction of a symmetric bivariate orthogonal polynomial sequence from a given sequence that is orthogonal to a weight function defined on the first quadrant of the plane. The Backlund-type matrix transformations of the involved three-term matrix coefficients play a crucial role in this description. Finally, we present a case study on the connections between classical orthogonal polynomials defined on the ball and those on the simplex.