MMN-2732
A note on sums of a class of series
Sungtae Jun; Gradimir V. Milovanovic; Insuk Kim; Arjun K. Rathie;Abstract
The aim of this note is to provide sums of a unified class of series of the form
\begin{equation*}
S_i(a)=\sum_{k=0}^{\infty} (-1)^{k} \binom{a-i}{k} \frac{1}{2^{k}(a+k+1)}
\end{equation*}
in the most general form for any $i\in\ZZ$. For each $\nu\in\NN$, in four cases when $i=\pm 2\nu$ and $i=\pm(2\nu-1)$, simple explicit expressions for $S_i(a)$ are obtained, e.g.
\[S_{2\nu}(a)=\frac{2^{2\nu-1-a}}{(a-2\nu+1)_\nu}\left[\frac{\sqrt{\pi}\, \Gamma (a+1)}{\Gamma \left(a+\frac{3}{2}-\nu\right)}-P_{\nu-1}(a)\right],\]
where $P_\nu(a)$ is an algebraic polynomial in $a$ of degree $\nu$.
For $i=1$ and $a=n$ $(\in \mathbb N)$, we recover the well known sum of the series due to Vowe and Seiffert.
Several other known results due to Srivastava and Kim {\it et al.} can be considered as special cases of our result.
Vol. 20 (2019), No. 2, pp. 985-996