---
tag: W8VE25R5A7
kind: concept
name: Hypergeometric differential equation
msc: ["33C05", "34M03"]
specializes: [ZBVH2YE50B]
sources:
  - url: https://en.wikipedia.org/wiki/Hypergeometric_function
---

## Summary

The hypergeometric differential equation is the linear equation $z(1-z)w'' + [c - (a+b+1)z]w' - ab\,w = 0$.
It is a [[Fuchsian equation|ZBVH2YE50B]] with exactly three [[regular singular points|H96HXG6CA0]], at $0$, $1$ and $\infty$.
The [[hypergeometric function|463X5V31D9]] is its solution that is holomorphic at $0$ with value $1$ there.
Every second-order Fuchsian equation with three singular points reduces to this form.
One uses a Möbius change of variable and a change of the unknown function.

## Explanations

### L1 human

Many equations of physics, after separating variables, become second-order linear equations whose coefficients blow up at a few points.
The hypergeometric equation is the simplest interesting case: it has three such points, $0$, $1$ and $\infty$, and near each of them solutions behave like powers.
Solving it by a power series around $0$ gives the hypergeometric series.
For example, with $a = 1$, $b = c$ the equation is solved by $(1 - z)^{-1}$, which is the geometric series.

### L2 human

The local exponents are $0, 1 - c$ at $z = 0$; $0, c - a - b$ at $z = 1$; and $a, b$ at $z = \infty$.
Their sum is $1$, as Fuchs's relation requires for three singular points.
Studying how the solutions change around the singular points leads to the [[monodromy of the hypergeometric equation|T8SBE28DM1]].