Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ We admit that, in case of existence of all quantities present in the following expression, $$F_{a,b,c}(1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}.$$ Let $(u, v) \in \mathbb{N}^2$ such that $N \leqslant \min(u, v)$. By taking $a = -u$ and $c = v - N + 1$, show Vandermonde's identity: $$\binom{u+v}{N} = \sum_{k=0}^{N} \binom{u}{k} \binom{v}{N-k}.$$
Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by
$$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$
We admit that, in case of existence of all quantities present in the following expression,
$$F_{a,b,c}(1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}.$$
Let $(u, v) \in \mathbb{N}^2$ such that $N \leqslant \min(u, v)$. By taking $a = -u$ and $c = v - N + 1$, show Vandermonde's identity:
$$\binom{u+v}{N} = \sum_{k=0}^{N} \binom{u}{k} \binom{v}{N-k}.$$