Generalized Vandermonde determinants and characterization of divisibility sequences

We present a different proof of the characterization of non-degenerate linear recurrence sequences, which are also divisibility sequences, given by Van der Poorten, Bézivin, and Pethö in their paper “A Full Characterization of Divisibility Sequences” [1]. Our proof is based on an interesting determinant identity, related to impulse sequences, obtained 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 [1], in the general case of non-degenerate divisibility sequences having minimal polynomial with multiple roots.