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arXiv · 2507.17694

Bivariate multiple orthogonal polynomials of mixed type on the step-line

Manuel Mañas·Miguel Rojas·Jianwen Wu
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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-Pi\~neiro multiple orthogonal polynomials of mixed type on the triangle are computed by means of an LU factorization implemented in a dedicated Maple script.

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BibTeXRIS

Manuel Mañas, Miguel Rojas, Jianwen Wu. 2025-07-23. Bivariate multiple orthogonal polynomials of mixed type on the step-line. https://arxiv.org/abs/2507.17694

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