The Vandermonde identity is given by
$ \left(\begin{matrix} m + n \\ j \end{matrix}\right) = \displaystyle\sum_{j=0}^k \left(\begin{matrix} m \\ j \end{matrix}\right)\left(\begin{matrix} n \\ k-j \end{matrix}\right)$
or equivalently
$ \left(\begin{matrix} m + n \\ r+s \end{matrix}\right) = \displaystyle\sum_j \left(\begin{matrix} m \\ r+j \end{matrix}\right)\left(\begin{matrix} n \\ s-j \end{matrix}\right)$ where $j$ runs through both negative and positive integers.
I would like to know if there is any equivalent Vandermonde identity for the following sum, and if so, what would be the best way to derive it?
$ \displaystyle\sum_{j=0}^{\min\{s,m-r\}} \left(\begin{matrix} m \\ r+j \end{matrix}\right)\left(\begin{matrix} n \\ s-j \end{matrix}\right)$
Thanks for any help in advance!