I have a question about combinatorics/gamma functions. I would like some help with either disproving or proving the following statement:
Given $k\in\mathbb{N}, m_{1},...,m_{n}\in\mathbb{N}$ such that $m_{1}+...+m_{n}=k$ and $a_{1},...,a_{n} $ positive real numbers and $r_{1},...,r_{n}$ also positive real numbers.
Statement:
$\begin{eqnarray}\underset{m_{1}+...+m_{n}=k}{\sum}&\frac{\overset{n}{\underset{i=1}{\prod}}\left(\begin{array}{c} m_{i}+r_{i}\\ m_{i}\end{array}\right)a_{i}^{m_{i}}}{\left(\begin{array}{c} \underset{i=1}{\overset{n}{\sum}}r_{i}+k-1\\ k\end{array}\right)}=\frac{\left(a_{1}r_{1}+...+a_{n}r_{n}\right)^{k}}{\left(\underset{i=1}{\overset{n}{\sum}}r_{i}\right)^{k}}, \end{eqnarray}$ where the sum is taken over all natural numbers such that $m_{1} + ... + m_{n}=k$.