Irrationality of certain fast converging series and infinite products

Abstract

Let $(u_{n})_{n\geq0}$ be an unbounded sequence of positive integers such that $u_{n+1}=\alpha u_{n}^{2}+O(u_{n}^{\gamma})$ for some positive rational number $\alpha$ and some $\gamma\in\left] 0,2\right[ .$ Let $(r_{n})_{n\geq0}$ be a sequence of rational numbers satisfying ``weak'' growth conditions. We give necessary and sufficient conditions for the series $\sum_{n=0}^{\infty}r_{n}/u_{n}$ and the infinite product $\prod_{n=0}^{\infty}\left(1+r_{n}/u_{n}\right)$ to be rational numbers. Moreover, in case of irrationality, we obtain an upper bound for their irrationality exponents.