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arXiv · 2601.19416

A Rodrigues Formula for Multiple Orthogonal Polynomials on the Simplex

Lidia Fernández·Ana Foulquié-Moreno·Juan Antonio Villegas
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Abstract

Rodrigues formulas play a central role in the construction of orthogonal polynomials associated with classical measures. In this paper, we introduce a family of multivariate multiple orthogonal polynomials on the simplex obtained through a Rodrigues-type construction that combines features of bivariate Jacobi polynomials on the simplex and univariate Jacobi--Pi\~neiro polynomials. We prove that the proposed construction produces polynomials and establish their main structural properties, including symmetry and multiple orthogonality with respect to several measures. This yields a natural multivariate extension of the classical Jacobi--Pi\~neiro family and, to the best of our knowledge, provides one of the first Rodrigues-type constructions for multivariate multiple orthogonal polynomials. Furthermore, we formulate a bivariate Hermite--Pad\'e-type approximation problem and show that the proposed polynomial family naturally appears as a common denominator of the corresponding approximants. Numerical experiments illustrating the performance of the resulting approximations are also presented.

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Lidia Fernández, Ana Foulquié-Moreno, Juan Antonio Villegas. 2026-01-27. A Rodrigues Formula for Multiple Orthogonal Polynomials on the Simplex. https://arxiv.org/abs/2601.19416

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