Trying to extend Vandermonde's formula to the case where m and n are not integers I am currently trying to extend Vandermonde's formula of $\sum_j\binom{m}{j}\binom{n}{k-j} = \binom{m+n}{k}$ for $m,n,k$ to the point where $m$ and $n$ are not integers and I'm not sure how to tackle this problem.

The identity
$$\sum_{j+k=\ell} {n \choose j} {m \choose k} = {n+m \choose \ell}$$
is a polynomial identity in $n, m$ in two variables. For fixed $\ell$, Vandermonde tells you that it's true for all non-negative values of $n, m$. You can show from here that the LHS and RHS must in fact be identical polynomials (they have the same coefficients), so the identity holds for all values of $n, m$ (complex, for example). 
