Counterexamples to the conjectured transcendence of $\,\sum{1/(n+\alpha)^{k}}$, its closed-form summation and extensions to polygamma functions and zeta series

TL;DR

This paper disproves a recent conjecture that certain infinite series are transcendental, identifies specific counterexamples, and provides a closed-form expression for their sum, extending results to polygamma and zeta series.

Contribution

It shows counterexamples to a conjecture about the transcendence of specific series and derives a closed-form expression, clarifying their arithmetic nature and extending to related functions.

Findings

Counterexamples for odd k and half-integer alpha disprove the conjecture.

A closed-form expression for the series sum is derived.

Results extend to polygamma functions and zeta series.

Abstract

In a recent work, Gun and co-workers have proposed that $\,\sum_{n=-\infty}^{\infty}{(n+\alpha)^{-k}}\,$ is a transcendental number for all integer $\,k$, $k > 1$, and $\,\alpha \in \mathbb{Q} \backslash \mathbb{Z}$. Here in this work, this proposition is shown to be \emph{false} whenever $\,k\,$ is odd and $\,\alpha\,$ is a half-integer. It is also shown that these are the only counterexamples, which allows for a correct reformulation of the original proposition. This leads to a theorem yielding a closed-form expression for the summation of that series, which determines its arithmetic nature. The result is then extended to a sum of polygamma functions and some related zeta series. In view of the recurrent appearance of these series and functions in different areas of mathematics and applications, the closed-form results put forward here could well be included in modern computer algebra systems (CAS).