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

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\begin{align}
\sum_{m \geq 0}{mr^{m}\Gamma\pars{m + c} \over \Gamma\pars{m + 1}} & =
\pars{c - 1}!\,r\,\totald{}{r}
\sum_{m \geq 0}{\pars{m + c - 1}! \over m!\pars{c - 1}!}\,r^{m} =
\pars{c - 1}!\,r\,\totald{}{r}
\sum_{m \geq 0}{m + c - 1 \choose m}\,r^{m}
\\[5mm] & =
\pars{c - 1}!\,r\,\totald{}{r}
\sum_{m \geq 0}{-c \choose m}\pars{-1}^{m}\,r^{m} =
\pars{c - 1}!\,r\,\totald{}{r}\pars{1 - r}^{-c}
\\[5mm] & =
\pars{c - 1}!\,r\,\bracks{-c\pars{1 - r}^{-c - 1}\pars{-1}} =
\bbx{c!\,{r \over \pars{1 - r}^{c + 1}}}
\end{align}
