How to find a closed form expression for the following summation? How to find a closed form expression for the following summation?
$$ \sum_{m\geq 0} \frac{m r^m \Gamma(m+c)}{\Gamma(m+1)} $$

Since $\Gamma(m+c)=\int_{0}^{+\infty}t^{m+c-1}e^{-t}\,dt$, for any $r<1$ the given series can be written as
$$ \int_{0}^{+\infty}\sum_{m\geq 0}\frac{m r^m t^{m+c-1}}{m!} e^{-t}\,dt =\int_{0}^{+\infty} r t^c e^{(r-1)t}\,dt=\color{red}{\frac{r\,\Gamma(c+1)}{(1-r)^{c+1}}}.$$
