A Rodrigues Formula for Multiple Orthogonal Polynomials on the Simplex

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--Piñeiro 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--Piñeiro 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--Padé-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.