Jacobi-Pi\~neiro Multiple Orthogonal Polynomials on the simplex

TL;DR

This paper introduces a Rodrigues formula-based method for constructing multivariate Jacobi-Piñeiro polynomials on the simplex, extending classical univariate polynomials to higher dimensions and applying them to the Hermite-Padé approximation problem.

Contribution

It provides a Rodrigues formula for multivariate Jacobi-Piñeiro polynomials and demonstrates their application to the bivariate Hermite-Padé problem.

Findings

Explicit Rodrigues formula for multivariate Jacobi-Piñeiro polynomials

Extension of classical polynomials to multivariate setting

Application to Hermite-Padé approximation on the triangle

Abstract

It is known that Rodrigues formulas provide a very powerful tool to compute orthogonal polynomials with respect to classical weights. We provide an example of bivariate multiple polynomials on the simplex defined via a Rodrigues formula. This approach offers a natural generalization of Jacobi--Pi\~neiro polynomials to the multivariate setting. Moreover, we apply these polynomials to the study of the bivariate Hermite--Pad\'e problem on the triangle.