## The irrationality of certain infinite products.(English)Zbl 0625.10027

The author considers a convergent product $$\prod^{\infty}_{n=1}(1+b_ n/a_ n)$$ with $$a_ n$$, $$b_ n$$ positive integers, and he gives a condition to have the product irrational. For example, given a positive integer k, $$\prod^{\infty}_{n=1}(1+k/a_ n)$$ is irrational provided the inequality $$a_{n+1}>a^ 2_ n+(k-1)a_ n+1-k$$ is true for n large enough.
This result extends a theorem of G. Cantor (1932) and also a result due to I. Sándor. He also notices connections with a false assertion made by Froda on V. Brun’s irrationality criterion [Arch. f. Math. og Nat. 31, No. 3, 6 p. (1910; Zbl 1234.11092)].
Reviewer: A.Escassut

### MSC:

 11J81 Transcendence (general theory)

Zbl 1234.11092