I know the addition theorem for binomial coefficients:
$\binom{x^*+y^*}{n}=\sum_{k=0}^n \binom{x^*}{n-k} \binom{y^*}{k}$
for $x^*,y^*\in \mathbb{R}$ and $n \in \mathbb{N}$.
Using this theorem i can prove identities such
$\binom{2n}{n}=\sum_{k=0}^n \binom{n}{k}^2$ easily by applying $x^*=y^*=n$.
But what about the identity:
$\binom{x+y+n-1}{n}=\sum_{k=0}^n \binom{x+n-k-1}{n-k} \binom{y+k-1}{k}$
Of course we see that it results from the Theorem but is it a problem that $k$ is "part of $x^*$ and $y^*$ here?
