Generalized Vandermonde determinants and characterization of divisibility 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 Pethö in their paper [1]. 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 [1], in the general case of non-degenerate divisibility sequences having minimal polynomial with multiple roots.

Classifying linear division sequences

“…by Proposition 2, which is the main theorem of Bézivin et al [5]. (Our proof is a modification of Barbero [3], which is substantially easier than the proof in [5]. )…”

“…If there are characteristic roots α i = α j such that α i /α j is a root of unity then (u n ) n≥0 is degenerate. 3 Much more important for our considerations is the period M of the linear recurrence sequence (u n ) n≥0 which is defined to be the smallest positive integer such that whenever α e 1 1 • • • α em m (each e i ∈ Z) is a root of unity, it is an Mth root of unity. 4 Thus if (u n ) n≥0 is degenerate then M > 1.…”

Linear divisibility sequences and Salem numbers

“…See [2] and [4]. In other words, there exist a sequence c = (c n ) ∞ n=0 such that b n = a n c n , for any index n.…”

Characterization of the strong divisibility property for generalized Fibonacci polynomials

“…We now prove that S(2) is also true. It is easy to see that gcd 2 . This implies that r(x) divides α 2 g(x)(G * k (x)) 2 .…”

“…In 2005 Rayes, et al [24] proved that the strong divisibility property holds partially for the Chebyshev polynomials (we prove the general result in Theorem 5.6). Over the years several other authors [2,3,8,16,17,18,20,21,22,23,25] have also been interested in the divisibility properties of sequences.…”