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New strong divisibility sequences. (English) Zbl 1512.11016

A sequence of any integer numbers \(\{a_n\}\) is said to be a divisibility sequence if \(a_m\mid a_n\), whenever \(m\mid n\), and is called a strong divisibility sequence if \(\gcd(a_m,a_n)=a_{\gcd(m,n)}\).
The strong divisibility sequences and their weaker version have been studied for more than one century. The Fibonacci numbers \(\{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, \dots\}\) are perhaps the best known non-trivial strong divisibility sequence.
In the paper under review, the authors establish the periodic continuants by using Chebyshev polynomials of the second kind. Then, from \(\{f_n\}\) they, under certain conditions, generate new strong divisibility sequences. At the same time, the authors recover the connection between the sequences defined by recurrence relations with two terms and the determinants of tridiagonal matrices. This is effective in the spirit of some ideas proposed by Édouard Lucas back in 1878.
Namely, the authors provide new families of divisibility and strong divisibility sequences based on some factorization properties of Chebyshev polynomials in this paper.

MSC:

11B37 Recurrences
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B83 Special sequences and polynomials
15B05 Toeplitz, Cauchy, and related matrices
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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