Hypergeometric series

Summary

The Gauss hypergeometric series is the power series $\sum_{n \ge 0} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$, written ${}_2F_1(a, b; c; z)$, where $(x)_n$ is the rising factorial. The ratio of consecutive coefficients is a rational function of $n$, of degree $2$ over degree $2$. Many elementary functions are special cases, and its sum on the unit disc is the starting point of the hypergeometric function.