Extended homogeneous bivariate orthogonal polynomials: symbolic and numerical Gaussian cubature formula

An extension of the bivariate homogeneous orthogonal polynomials can be introduced by using a linear functional with complex moments obtained from the series expansions of a bivariate function at the origin and infinity. They are used to construct the bivariate homogeneous two-point Padé approximant and to solve related problems. In this paper we study the connection between extended homogeneous bivariate orthogonal polynomials and symbolic Gaussian cubature formula for the approximation of bivariate integrals over domains with non-negative weight functions. By extension to the two-point case, a new symbolic Gaussian cubature is presented. A new numerical cubature is also developed. Finally, some numerical examples are given to illustrate our results.