On the transcendence of some infinite sums

TL;DR

This paper extends the understanding of when certain infinite sums involving rational functions are transcendental, providing new necessary and sufficient conditions for degree 4 polynomials.

Contribution

It generalizes previous results by establishing conditions for the transcendence of sums with degree 4 polynomials, building on earlier work for degree 3.

Findings

Provides necessary and sufficient conditions for transcendence when degree of Q is 4

Extends previous results from degree 3 to degree 4

Focuses on sums with simple rational zeros of Q(x)

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.