Suites récurrentes linéaires. Propriétés algébriques et arithmétiques. (Linear recurrent sequences. Algebraic and arithmetic properties). (French) Zbl 0626.10008

Cerlienco, L.; Mignotte, M.; Piras, F.

Enseign. Math., II. Sér. 33, 67-108 (1987).

This paper is an overview of properties of linear recurrent sequences which, as the authors remark, does not pretend to be complete. The first part of the paper is devoted to algebraic properties such as their generating functions, which are rational, and a very formal interpretation in terms of bialgebras. The reader, including the reviewer, may have some reservations about such a fargoing formalization of such an elementary object as recurrences. Common sense usually works better. As application the authors give some algorithms to manipulate polynomials in intricate ways.
The second, arithmetic, part contains a summary of known facts on multiplicities of values, diophantine equations and related topics. Particularly nice are the proofs on the squares in the Fibonacci sequence and the solution of \(\left( \begin{matrix} x\\ 2\end{matrix} \right)=3\cdot 2^ k-5\).

Reviewer: F.Beukers

MSC:

11B37	Recurrences
11-02	Research exposition (monographs, survey articles) pertaining to number theory

Keywords:

arithmetic properties; overview; linear recurrent sequences; algebraic properties; generating functions; bialgebras; multiplicities of values; diophantine equations; squares in the Fibonacci sequence