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

MSC:
11J81 Transcendence (general theory)
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