Geronimus transformations for sequences of $d$-orthogonal polynomials

TL;DR

This paper extends Geronimus transformations to sequences of d-orthogonal polynomials, analyzing the resulting sequences, their relationships, and associated Hessenberg matrices to deepen understanding of their structure.

Contribution

It introduces a novel extension of Geronimus transformations specifically for d-orthogonal polynomial sequences and explores their properties and matrix representations.

Findings

New relationships between transformed polynomial sequences

Analysis of Hessenberg matrices associated with the transformations

Insights into the structure of d-orthogonal polynomials

Abstract

In this paper an extension of the concept of Geronimus transformation for sequences of $d$-orthogonal polynomials $\{P_n\}$ is introduced. The transformed sequences $\{P^{(k)}_n\}$, for $ k=1,\ldots, d,$ are analyzed and some relationships between these new sequences of polynomials are given. Also the associated Hessenberg matrices and their transformed are studied.