Bivariate multiple orthogonal polynomials of mixed type on the step-line
Manuel Mañas, Miguel Rojas, Jianwen Wu
TL;DR
This work develops a comprehensive framework for mixed-type, bivariate multiple orthogonal polynomials along the step-line through Gauss–Borel LU factorization of a carefully structured moment matrix. It introduces monomial matrices, graded lexicographic ordering, and shift operators to define biorthogonal polynomial families and their orthogonality relations, then derives recurrence relations and Christoffel–Darboux-type kernels with an ABC-type theorem in this setting. The approach is illustrated via a detailed bivariate Jacobi–Piñeiro example on the triangle, implemented with Maple, highlighting explicit polynomials and biorthogonality proofs. By connecting moment-factorization structures to spectral-like recurrences and integral kernels, the paper advances multivariate mixed-type orthogonality and its potential links to quadrature, integrable systems, and numerical analysis.
Abstract
This article studies bivariate multiple orthogonal polynomials of the mixed type on the step-line. The analysis is based on the LU factorization of a moment matrix specifically adapted to this framework. The orthogonality and biorthogonality relations satisfied by these polynomials are identified, and their precise multi-degrees are determined. The corresponding recurrence relations and the growing band matrices that encode them are also derived. Christoffel-Darboux kernels and the associated Christoffel-Darboux-type formulas are obtained. An ABC-type theorem is established, relating the inverse of the truncated moment matrix to these kernels. As an illustration, the bivariate Jacobi-Piñeiro multiple orthogonal polynomials of mixed type on the triangle are computed by means of an LU factorization implemented in a dedicated Maple script.