Recurrence formulas for multivariate orthogonal polynomials

Abstract

In this paper, necessary and sufficient conditions are given so that multivariate orthogonal polynomials can be generated by a recurrence formula. As a consequence, orthogonal polynomials of total degree n in d variables that have dim ⁡ Π n d \dim \Pi _n^d common zeros can now be constructed recursively. The result is important to the construction of Gaussian cubature formulas.