Multiple orthogonal polynomials: Pearson equations and Christoffel formulas

TL;DR

This paper explores the theory of multiple orthogonal polynomials, establishing connections between spectral matrices, Pearson equations, and Christoffel transformations, with detailed case studies and new relations for polynomial modifications.

Contribution

It introduces new connections between spectral matrices, Pearson equations, and Christoffel transformations for multiple orthogonal polynomials, including detailed case studies and symmetry relations.

Findings

Derived differential equations for multiple orthogonal polynomials.

Established connections between different polynomial families via Christoffel transformations.

Analyzed the Jacobi-Piñeiro case to illustrate symmetries and parameter shifts.

Abstract

Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss-Borel factorization of the moment matrix is given. It is shown a hidden freedom exhibited by the spectral system related to the multiple orthogonal polynomials. Pearson equations are discussed, a Laguerre-Freud matrix is considered, and differential equations for type I and II multiple orthogonal polynomials, as well as for the corresponding linear forms are given. The Jacobi-Pi\~neiro multiple orthogonal polynomials of type I and type II are used as an illustrating case and the corresponding differential relations are presented. A permuting Christoffel transformation is discussed, finding the connection between the different families of multiple orthogonal polynomials. The Jacobi-Pi\~neiro case provides a convenient illustration of these symmetries, giving linear relations between different polynomials with shifted and permuted parameters. We also present the general theory for the perturbation of each weight by a different polynomial or rational function aka called Christoffel and Geronimus transformations. The connections formulas between the type II multiple orthogonal polynomials, the type I linear forms, as well as the vector Stieltjes-Markov vector functions is also presented. We illustrate these findings by analyzing the special case of modification by an even polynomial.