On the transcendence of some infinite sums

Abstract

In this paper we investigate the infinite convergent sum $T={\sum }_{n=0}^{\infty }\frac{P(n)}{Q(n)}$, where $P(x)\in \bar{\mathbb{Q}}[x]$, $Q(x)\in \mathbb{Q}[x]$ and $Q(x)$ has only simple rational zeros. N. Saradha and R. Tijdeman have obtained sufficient and necessary conditions for the transcendence of $T$ if the degree of $Q(x)$ is 3. In this paper we give sufficient and necessary conditions for the transcendence of $T$ if the degree of $Q(x)$ is 4 and $Q(x)$ is reduced.

arXiv comment15pages