---
tag: Y4759V8CY6
kind: statement
role: theorem
name: Gauss multiplication formula
msc: [33B15]
uses: [0V3FZMVB4K, 0NSD1TPHR0]
about: [0H5SYBSED9]
sources:
  - url: https://en.wikipedia.org/wiki/Multiplication_theorem
---

## Statement

Let $\Gamma$ be the [[gamma function|0V3FZMVB4K]], extended to a [[meromorphic function|0NSD1TPHR0]] on $\C$, and let $m$ be a positive integer.
For every $z \in \C$ with $mz \notin \set{0, -1, -2, \ldots}$,
$$\prod_{k=0}^{m-1} \Gamma\!\left(z + \frac{k}{m}\right) = (2\pi)^{(m-1)/2} \, m^{1/2 - mz} \, \Gamma(mz),$$
where $m^{1/2 - mz} = e^{(1/2 - mz)\ln m}$.

## Remarks

For $x > 0$, the function $f(x) = \frac{m^{x - 1/2}}{(2\pi)^{(m-1)/2}} \prod_{k=0}^{m-1} \Gamma\!\left(\frac{x + k}{m}\right)$ satisfies $f(x+1) = x f(x)$ and $\ln f$ is convex.
So $f = c\,\Gamma$ for a constant $c$, by the argument of the [[Bohr–Mollerup theorem|QFF87AXWN2]]; the constant is found from $f(1) = 1$, which uses $\prod_{k=1}^{m-1}\Gamma(k/m) = (2\pi)^{(m-1)/2}/\sqrt{m}$, a consequence of the [[reflection formula|WF7KQAQTS2]].
Alternatively, the constant follows from [[Stirling's formula|53750WSJ6K]] as $z \to \infty$.
For $m = 1$ the formula is trivial, and for $m = 2$ it is the [[Legendre duplication formula|CND4E1PY9V]], since $(2\pi)^{1/2} 2^{1/2 - 2z} = 2^{1-2z}\sqrt{\pi}$.

## Explanations

### L1 human

The multiplication formula splits $\Gamma$ at $m$ times a number into a product of $m$ values of $\Gamma$ at evenly spaced points.
It generalizes the duplication formula from $2$ to any whole number $m$.

### L2 human

Taking logarithms and differentiating gives the matching multiplication formula for the [[digamma function|HWRRXB0YXS]].
Similar multiplication theorems hold for other special functions, such as the Hurwitz zeta function and the Bernoulli polynomials.
Setting $z = 1/m$ gives the product $\prod_{k=1}^{m-1}\Gamma(k/m) = (2\pi)^{(m-1)/2}/\sqrt{m}$.