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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
11J81 Transcendence (general theory)