Multiple Orthogonal Polynomials of two real variables

TL;DR

This paper introduces a generalization of Multiple Orthogonal Polynomials to two variables, extending their theoretical framework and providing initial examples, which broadens their applicability in mathematical analysis.

Contribution

The paper develops the theory of Multiple Orthogonal Polynomials in two variables, including key properties and examples, filling a gap in multivariate orthogonal polynomial research.

Findings

Extended properties of multivariate orthogonal polynomials

Examples illustrating the new two-variable polynomials

Foundation for future applications in approximation and analysis

Abstract

Polynomials known as Multiple Orthogonal Polynomials in a single variable are polynomials that satisfy orthogonality conditions concerning multiple measures and play a significant role in several applications such as Hermite-Pad\'e approximation, random matrix theory or integrable systems. However, this theory has only been studied in the univariate case. We give a generalization of Multiple Orthogonal Polynomials for two variables. Moreover, an extended version of some of the main properties are given. Additionally, some examples are given along the paper.