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Irrationalité de la somme des inverses de certaines suites récurrentes. (Irrationality of the sum of reciprocals in certain recurrence generated sequences). (French) Zbl 0682.10025

The author proves the irrationality of series \(\sum^{\infty}_{n=1}x^ n/w_ n\) where \(w_ n/x^ n\) is a Fibonacci sequence in applying processes due to R. Apéry. In other words, let \(r\in {\mathbb{Z}}^*\), \(s=\pm 1\) be such that \(r^ 2-4s>0\) and let \(\alpha\), \(\beta\) be the zeros of \(X^ 2-rX+s\). Let \(w_ n\) be a sequence in \({\mathbb{Z}}\) satisfying the recurrence relation \(w_ n=rw_{n- 1}-sw_{n-2}\) with \(w_ n\neq 0\) whenever \(n\geq 1\). Then there exist \(c_ 1\) and \(c_ 2\in {\mathbb{R}}\) such that \(w_ n=c_ 1\alpha^ n+c_ 2\beta^ n\). If \(c_ 1c_ 2\neq 0\), then the author shows that for every \(x\in {\mathbb{Z}}\) such that \(| x| <| \alpha |\) and \(| c_ 1c_ 2| x^ 2<| \alpha |\) the sum \(\theta\) of the series \(\sum^{\infty}_{n=1}x^ n/w_ n\) is irrational.
The author first constructs a sequence \((p_{n,k},q_{n,k})\) so that \(p_{n,k}/q_{n,k}\) converges to \(\theta\) uniformly with respect to k and then the sequence \(p_{n,n}/q_{n,n}\) quickly converges to \(\theta\), so quickly that \(\theta\) is irrational.
Reviewer: A.Escassut

MSC:

11J81 Transcendence (general theory)
11B37 Recurrences
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