On the construction of a family of transcendental valued infinite products. (English) Zbl 1062.11048

Nyblom, M. A.

Fibonacci Q. 42, No. 4, 353-358 (2004).

The main result of this paper is the following:
Theorem: Suppose \(\{a_n\}_{n=1}^\infty\) is a sequence of positive integers greater than unity and such that, for a fixed \(K>2\) \[ \liminf_{n\to \infty}\frac{a_{n+1}}{a_n^{K+1}}>2, \] then the infinite product \(\prod_{n=1}^\infty (1+\frac 1{a_n})\) converges to a transcendental number.
The proof is based on Roth’s theorem. Some examples involving Fibonacci and Lucas sequences are included.

Reviewer: Jaroslav Hančl (Ostrava)

MSC:

11J81

Transcendence (general theory)

Keywords:

infinite product; transcendence