Generalized Vandermonde Determinants and Characterization of Divisibility Sequences

TL;DR

This paper offers a new proof for the characterization of non-degenerate divisibility sequences using a generalized Vandermonde determinant, providing more precise results especially for sequences with multiple roots.

Contribution

It introduces a novel determinant-based proof for the characterization of divisibility sequences, extending the results to cases with multiple roots in the minimal polynomial.

Findings

New proof of divisibility sequence characterization

More precise form for the resultant sequence with multiple roots

Extension of previous results to broader class of sequences

Abstract

We present a different proof of the characterization of non--degenerate recurrence sequences, which are also divisibility sequences, given by Van der Poorten, Bezevin, and Petho in their paper "A Full Characterisation of Divisibility Sequence". Our proof is based on an interesting determinant identity related to impulse sequences, arising from the evaluation of a generalized Vandermonde determinant. As a consequence of this new proof we can find a more precise form for the resultant sequence presented in their paper, in the general case of non--degenerate divisibility sequences having minimal polynomial with multiple roots.