Quadratic transformation and matrix biorthogonal polynomials: an $\mathcal{LU}$ factorization approach

TL;DR

This paper introduces an LU factorization method for matrix biorthogonal polynomials under specific zero-entry conditions in the Gram matrix, exploring quadratic transformations and related kernel polynomial representations.

Contribution

It develops a novel LU approach for biorthogonal polynomials with zero entries in the Gram matrix, including quadratic transformations and the ABC Theorem for kernel polynomial representations.

Findings

LU factorization for special Gram matrices

Analysis of quadratic transformations in polynomial theory

Representation of kernel polynomials under Hankel symmetry

Abstract

The manuscript presents the $LU$ approach to matrix biorthogonal polynomials when all the even ordered entries in the Gram matrix are zero. This arises in case of a quadratic transformation which is briefly discussed. Further, the main diagonal of the Gram matrix is a zero diagonal and we present the theory that follows from this fact. Precisely, we discuss the Christoffel transformation and matrix representations of the kernel polynomials, usually called the ABC Theorem. Finally, we provide an illustration of our results assuming the Gram matrix has Hankel symmetry.