Rational symbolic cubature rules over the first quadrant in a Cartesian plane

[摘要] In this paper we introduce a new symbolic Gaussian formula for the evaluation of an integral over the first quadrant in a Cartesian plane, in particular with respect to the weight function $w(x)=\exp(-x^T x-1/x^T x)$, where $x=(x_1,x_2)^T\in \mathbb{R}^2_+$. It integrates exactly a class of homogeneous Laurent polynomials with coefficients in the commutative field of rational functions in two variables. It is derived using the connection between orthogonal polynomials, two-point Padé approximants, and Gaussian cubatures. We also discuss the connection to two-point Padé-type approximants in order to establish symbolic cubature formulas of interpolatory type. Numerical examples are presented to illustrate the different formulas developed in the paper.

[学科分类] 数学（综合）

[关键词] homogeneous orthogonal polynomials;homogeneous two-point Padé;symbolic Gaussian cubature