---
tag: M2V5A38W5C
kind: statement
role: theorem
name: Clausen's formula
msc: [33C20, 33C05]
uses: [463X5V31D9, VDBAW9PN88]
about: [463X5V31D9, VDBAW9PN88]
sources:
  - url: https://en.wikipedia.org/wiki/Clausen%27s_formula
same_as:
  wikidata: Q5129539
  wikipedia: Clausen's formula
---

## Statement

Let $a$ and $b$ be complex numbers such that neither $a + b + \frac12$ nor $2a + 2b$ lies in $\set{0, -1, -2, \dots}$.
Then for every complex $x$ with $\abs{x} < 1$,
$$\left[ {}_2F_1\left( a, b; a + b + \tfrac12; x \right) \right]^2 = {}_3F_2\left( \begin{matrix} 2a,\ 2b,\ a + b \\ a + b + \frac12,\ 2a + 2b \end{matrix} ; x \right),$$
where ${}_2F_1$ is the [[hypergeometric function|463X5V31D9]] and ${}_3F_2$ is the [[generalized hypergeometric series|VDBAW9PN88]].

## Remarks

One proof compares differential equations.
If $y$ solves a second order linear equation, then $y^2$ solves a third order linear equation, the symmetric square.
For the hypergeometric equation with $c = a + b + \frac12$, the symmetric square turns out to be the generalized hypergeometric equation of the ${}_3F_2$ on the right.
Both sides are analytic at $0$ with value $1$, and that equation has only one such solution, so they agree.
Check of the first coefficient: on the left, the coefficient of $x$ is $\frac{2ab}{a + b + \frac12}$.
On the right it is $\frac{2a \cdot 2b \cdot (a + b)}{(a + b + \frac12)(2a + 2b)} = \frac{2ab}{a + b + \frac12}$, as it should be.
For $a = b = \frac14$ the formula reads ${}_2F_1(\frac14, \frac14; 1; x)^2 = {}_3F_2(\frac12, \frac12, \frac12; 1, 1; x)$.
The relation $c = a + b + \frac12$ between the parameters is what makes the symmetric square equation hypergeometric, and the proof uses it.

## Explanations

### L1 human

Clausen's formula says that the square of a certain hypergeometric function is again a hypergeometric function, with one more parameter.
Squaring a power series normally mixes the coefficients in a complicated way.
Here the coefficients of the square come out as a simple ratio of rising factorials.

### L2 human

The formula links two levels of the hypergeometric hierarchy, ${}_2F_1$ and ${}_3F_2$.
It gives positivity results: when the ${}_2F_1$ is real, the ${}_3F_2$ on the right is a square, hence nonnegative.
It is also used to turn identities for squares of elliptic integrals into series for $1/\pi$.