Transcendence of Some Infinite Series

Abstract

In this chapter, we investigate the transcendental nature of the sum

$$\displaystyle{\mathop{\sum\nolimits^{\prime}}_{n\in \mathbb{Z}}{ A(n) \over B(n)}}$$

where A(x), B(x) are polynomials with algebraic coefficients with $\deg A <\deg B$ and the sum is over integers n which are not zeros of B(x). We relate this question to a conjecture originally due to Schneider. A stronger version of this conjecture was later suggested by Gel’fond and Schneider. In certain cases, these conjectures are known and this allows one to obtain some unconditional results of a general nature.