Revisiting Biorthogonal Polynomials. An $LU$ factorization discussion

TL;DR

This paper revisits the theory of biorthogonal polynomials through the lens of LU factorization of Gram matrices, providing new insights into their properties, classical examples, and perturbation formulas.

Contribution

It introduces an LU factorization framework for biorthogonal polynomials, connecting spectral matrices, Christoffel-Darboux kernels, and perturbation formulas.

Findings

LU factorization clarifies biorthogonal polynomial structure

Classical orthogonal polynomials are characterized within this scheme

Christoffel formulas for perturbations are derived using this approach

Abstract

The Gauss-Borel or $LU$ factorization of Gram matrices of bilinear forms is the pivotal element in the discussion of the theory of biorthogonal polynomials. The construction of biorthogonal families of polynomials and its second kind functions, of the spectral matrices modeling the multiplication by the independent variable $x$, the Christoffel-Darboux kernel and its projection properties, are discussed from this point of view. Then, the Hankel case is presented and different properties, specific of this case, as the three terms relations, Heine formulas, Gauss quadrature and the Christoffel-Darboux formula are given. The classical orthogonal polynomial of Hermite, Laguerre and Jacobi type are discussed and characterized within this scheme. Finally, it is shown who this approach is instrumental in the derivation of Christoffel formulas for general Christoffel and Geronimus perturbations of the bilinear forms.