---
tag: 428M453CR2
kind: concept
name: Hypergeometric series
msc: ["33C05"]
specializes: [0MZB06ST2B]
sources:
  - url: https://en.wikipedia.org/wiki/Hypergeometric_function
---

## Summary

The Gauss hypergeometric series is the [[power series|0MZB06ST2B]] $\sum_{n \ge 0} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$, 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|463X5V31D9]].

## Explanations

### L1 human

The geometric series $1 + z + z^2 + \cdots$ has the same ratio $z$ between all consecutive terms.
In the hypergeometric series the ratio of the term with $z^{n+1}$ to the term with $z^n$ is $\frac{(n+a)(n+b)}{(n+c)(n+1)} z$, which changes with $n$ but tends to $z$.
So the series converges for $\abs{z} < 1$, just like the geometric series.
With $a = 1$ and $b = c$ it is exactly the geometric series $1/(1 - z)$.
With $a = b = 1$ and $c = 2$ it gives $-\log(1 - z)/z$.