  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).
