## 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

### Keywords:

Apery’s method; irrationality; Fibonacci sequence