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

2025/07/23 by Mañas, Manuel, Rojas, Miguel, Wu, Jianwen
#33BXX #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2507.17694

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.

Citations

• Multiple Orthogonal Polynomials of two real variables
• General Geronimus Perturbations for Mixed Multiple Orthogonal Polynomials
• Integral and hypergeometric representations for multiple orthogonal polynomials
• Classical multiple orthogonal polynomials for arbitrary number of weights and their explicit representation
• Banded totally positive matrices and normality for mixed multiple orthogonal polynomials
• Hypergeometric Expressions for Type I Jacobi-Piñeiro Orthogonal Polynomials with Arbitrary Number of Weights
• Finite Markov chains and multiple orthogonal polynomials
• Markov Chains and Multiple Orthogonality
• Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials
• Positive bidiagonal factorization of tetradiagonal Hessenberg matrices
• Revisiting Biorthogonal Polynomials. An LU factorization discussion
• Orthogonal polynomials in and on a quadratic surface of revolution
• Multivariate orthogonal Laurent polynomials and integrable systems
• Generalized characteristic polynomials and Gaussian cubature rules
• Krall--type Orthogonal Polynomials in several variables
• Moment matrices and multi-component KP, with applications to random matrix theory

Related

• Bidiagonal factorization of recurrence banded matrices in mixed multiple orthogonality
• Mixed Multiple Orthogonal Laurent Polynomials on the Unit Circle
• Uvarov Perturbations for Multiple Orthogonal Polynomials of the Mixed Type on the Step-Line
• Applications of multiple orthogonal polynomials with hypergeometric moment generating functions
• Quadratic decomposition of bivariate orthogonal polynomials