---
tag: 463X5V31D9
kind: concept
name: Hypergeometric function
msc: ["33C05"]
specializes: [0NAHXR1FRH]
sources:
  - url: https://en.wikipedia.org/wiki/Hypergeometric_function
same_as:
  wikidata: Q21028472
  wikipedia: Hypergeometric function
---

## Summary

The Gauss hypergeometric function ${}_2F_1(a, b; c; z)$ is the sum of the [[hypergeometric series|428M453CR2]] on the unit disc.
It is extended by [[analytic continuation|0M3YQJ81NB]] to the cut plane $\C \setminus [1, \infty)$.
It solves the [[hypergeometric differential equation|W8VE25R5A7]], to which every second-order linear equation with three regular singular points can be reduced.
Most classical special functions, such as Legendre functions, complete elliptic integrals, Jacobi polynomials and many elementary functions, are special cases or limits of it.

## Explanations

### L1 human

The hypergeometric function is a single formula with three parameters $a, b, c$ that produces a large family of familiar functions.
For different choices of the parameters it gives powers $(1-z)^{-a}$, logarithms, inverse sines, and orthogonal polynomials.
The power series only converges for $\abs{z} < 1$, but the function can be extended to all complex $z$ except a cut along $[1, \infty)$.
For example, ${}_2F_1(\tfrac12, \tfrac12; \tfrac32; z^2) = \frac{\arcsin z}{z}$ for $0 < \abs{z} < 1$.
Because so many functions are special cases, a single identity for ${}_2F_1$ often gives many classical identities at once.