Hypergeometric series

Remarks

If $a$ or $b$ is a non-positive integer $-m$, then $(a)_n$ or $(b)_n$ vanishes for $n > m$ and the series is a polynomial of degree at most $m$. Otherwise the ratio test gives radius of convergence $1$. On the circle $\abs{z} = 1$ the series converges absolutely when $\operatorname{Re}(c - a - b) > 0$.

Example: for $a = b = 1$, $c = 2$, the coefficient is $\frac{n! \, n!}{(n+1)! \, n!} = \frac{1}{n+1}$, so ${}_2F_1(1, 1; 2; z) = \sum_n \frac{z^n}{n+1} = -\frac{\log(1 - z)}{z}$ for $0 < \abs{z} < 1$. Also ${}_2F_1(a, b; b; z) = \sum_n \frac{(a)_n}{n!} z^n = (1 - z)^{-a}$ by the binomial series.

Non-example: $c = 0$ is excluded, because $(0)_n = 0$ for every $n \ge 1$ and the coefficients are undefined. For $c = -m$ a negative integer the series is still meaningful when $a = -k$ with $0 \le k \le m$ and the sum is cut at $n = k$, but this needs a separate convention.

The notation $F(a, b; c; z)$ is also common.