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

arXiv comment9 pages, no figures. Last theorem is in a better format. Submitted (09/12/2017)