Gauss Multiplication Formula

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Theorem

Let $\Gamma$ denote the Gamma Function.


Then:

$\displaystyle \forall z \notin \left\{{-\frac m n: m \in \N}\right\}: \prod_{k \mathop = 0}^{n - 1} \Gamma \left({z + \frac k n}\right) = \left({2 \pi}\right)^{\left({n - 1}\right) / 2} n^{1/2 - n z} \Gamma \left({n z}\right)$

where $\N$ denotes the natural numbers.


Proof

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


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

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

$\blacksquare$


Source of Name

This entry was named for Carl Friedrich Gauss.


Sources