# Gauss Multiplication Formula

## Theorem

Let $\Gamma$ denote the Gamma Function.

Then:

$\displaystyle \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$

where $\N$ denotes the natural numbers.

## Proof

 $\displaystyle \map \Gamma {z + \frac k n}$ $=$ $\displaystyle \paren {z + \frac k n - 1} \map \Gamma {z + \frac k n - 1}$ Gamma Difference Equation $\displaystyle$ $=$ $\displaystyle \lim_{m \mathop \to \infty} \frac {m! m^{z + k / n - 1} } {\paren {z + \frac k n} \paren {z + \frac k n + 1} \cdots \paren {z + \frac k n - 1 + m} }$ Definition of Euler Form of Gamma Function $\displaystyle$ $=$ $\displaystyle \lim_{m \mathop \to \infty} \frac {\sqrt {2 \pi m} \paren {\frac m e}^m m^{z + k / n - 1} } {\paren {z + \frac k n} \paren {z + \frac k n + 1} \cdots \paren {z + \frac k n - 1 + m} }$ Stirling's Formula $\displaystyle$ $=$ $\displaystyle \lim_{m \mathop \to \infty} \frac {\sqrt {2 \pi} \paren {\frac {m n} e}^m m^{z + k / n - 1/2} } {\paren {n z + k} \paren {n z + k + n} \cdots \paren {n z + k - n + m n} }$

Taking the product for $k = 0$ to $n - 1$, we have:

 $\displaystyle \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n}$ $=$ $\displaystyle \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac {m n} e}^{m n} m^{n z - n / 2} m^{\sum_{k \mathop = 0}^{n - 1} k / n} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m n} }$ $\displaystyle$ $=$ $\displaystyle \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac {m n} e}^{m n} m^{n z - 1 / 2} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m n} }$ Sum of Arithmetic Sequence $\displaystyle$ $=$ $\displaystyle \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac m e}^m m^{n z - 1 / 2} n^{1 / 2 - n z} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }$ by substituting $m n \mapsto m$ $\displaystyle$ $=$ $\displaystyle \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^{n - 1} m! m^{n z - 1} n^{1 / 2 - n z} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }$ Stirling's Formula $\displaystyle$ $=$ $\displaystyle \paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \paren {n z - 1} \map \Gamma {n z - 1}$ Definition of Euler Form of Gamma Function $\displaystyle$ $=$ $\displaystyle \paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \map \Gamma {n z}$ Gamma Difference Equation

$\blacksquare$

## Source of Name

This entry was named for Carl Friedrich Gauss.