I am looking for relevant literature on sums of series involving the Gamma function. In particular, my interest is in series of the following form.
\begin{equation} \sum_{j=0}^\infty \frac{\Gamma(c(j+k)) h(j+k)}{\Gamma(n+c(j+k))j!} \end{equation}
For instance, when $h(j+k) = \Gamma(j+k-\sigma)$ and $c=1$, the sum is equal to the following \begin{equation} \Gamma(n-k+\sigma)\frac{\Gamma(k)\Gamma(k-\sigma)}{\Gamma(n)\Gamma(n+\sigma)} \end{equation}
Can someone suggest relevant literature for similar series, or can someone think of any obvious result for other values of $c$ and the function $h$?