Gauss's hypergeometric theorem

Remarks

When moreover $\operatorname{Re} c > \operatorname{Re} b > 0$, put $z = 1$ in Euler's integral representation: the integral becomes the beta integral $B(b, c - a - b)$, and the gamma factors simplify to the right side. Abel's theorem on power series identifies the value of the series at $1$ with the limit of the function as $z \to 1^-$. The general case follows by analytic continuation in the parameters, or from a contiguous relation.

Example: for $a = b = 1$, $c = 3$ the series is $\sum_{n \ge 0} \frac{n! \, n!}{(n+2)!/2 \cdot n!} = \sum_{n \ge 0} \frac{2}{(n+1)(n+2)} = 2$, and the right side is $\frac{\Gamma(3) \Gamma(1)}{\Gamma(2) \Gamma(2)} = 2$.

With $a = -n$ for an integer $n \ge 0$, the theorem gives the Chu–Vandermonde identity.

The condition $\operatorname{Re}(c - a - b) > 0$ is needed: for $a = b = 1$, $c = 2$ one has $c - a - b = 0$, and the series $\sum_n \frac{1}{n+1}$ diverges.

Going deeper

For a student

The theorem is the basic summation formula for ${}_2F_1$, and most closed-form evaluations of ${}_3F_2$ and higher series, such as those of Pfaff–Saalschütz and Dixon, are its relatives. It also gives the connection coefficients between the solutions at $0$ and at $1$ of the hypergeometric equation, since these are values of ${}_2F_1$ at $1$. The terminating case, Chu–Vandermonde, is a polynomial identity behind many binomial sums.