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 $N \in \mathbb{N}, c \in D, a \in \mathbb{R}$ such that $c - a \in D$. Justify the existence of $F_{a,-N,c}(1)$ and prove that $$\sum_{k=0}^{N} (-1)^k \binom{N}{k} \frac{[a]_k}{[c]_k} = \frac{[c-a]_N}{[c]_N}.$$
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 $N \in \mathbb{N}, c \in D, a \in \mathbb{R}$ such that $c - a \in D$. Justify the existence of $F_{a,-N,c}(1)$ and prove that
$$\sum_{k=0}^{N} (-1)^k \binom{N}{k} \frac{[a]_k}{[c]_k} = \frac{[c-a]_N}{[c]_N}.$$