Transcendence of the values of certain series with Hadamard's gaps

摘要

Transcendence of the number $ sum_{k=0}^infty alpha^{r_k} $ , where $ alpha $ is an algebraic number with 0 < $ midalphamid $ > 1 and $ {r_k}_{kgeqq0} $ is a sequence of positive integers such that $ lim_{ktoinfty}, r_{k+1}/r_k = d in mathbb{N}, backslash {1} $ , is proved by Mahler's method. This result implies the transcendence of the number $ sum_{k=0}^infty alpha^{kd^k} $ .