Gauss Multiplication Formula

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Theorem

Let $\Gamma$ denote the Gamma Function.

Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the non-zero natural numbers.

Then:

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


Proof

\(\ds \map \Gamma {z + \frac k n}\) \(=\) \(\ds \paren {z + \frac k n - 1} \map \Gamma {z + \frac k n - 1}\) Gamma Difference Equation
\(\ds \) \(=\) \(\ds \lim_{m \mathop \to \infty} \paren {z + \frac k n - 1} \frac {m! m^{z + k / n - 1} } {\paren {z + \frac k n - 1} \paren {z + \frac k n} \paren {z + \frac k n + 1} \paren {z + \frac k n + 2} \cdots \paren {z + \frac k n - 1 + m} }\) Definition of Euler Form of Gamma Function
\(\ds \) \(=\) \(\ds \lim_{m \mathop \to \infty} \frac {\sqrt {2 \pi m} \paren {\frac m e}^m m^{z + k / n - 1} } {\paren {z + \frac k n} \paren {z + \frac k n + 1} \paren {z + \frac k n + 2} \cdots \paren {z + \frac k n - 1 + m} }\) Stirling's Formula and canceling $\paren {z + \frac k n - 1}$
\(\ds \) \(=\) \(\ds \lim_{m \mathop \to \infty} \frac {\sqrt {2 \pi m} \paren {\frac m e}^m m^{z + k / n - 1} } {\paren {z + \frac k n} \paren {z + \frac k n + 1} \paren {z + \frac k n + 2} \cdots \paren {z + \frac k n - 1 + m} } \times \frac {n^m} {n^m}\) multiplying top and bottom by $n^m$
\(\ds \) \(=\) \(\ds \lim_{m \mathop \to \infty} \frac {\sqrt {2 \pi} \paren {\frac {m n} e}^m m^{z + k / n - 1/2} } {\paren {n z + k} \paren {n z + k + n} \paren {n z + k + 2n} \cdots \paren {n z + k - n + m n} }\)


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

\(\ds \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n}\) \(=\) \(\ds \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac {m n} e}^{m n} m^{n z - n / 2} m^{\sum_{k \mathop = 0}^{n - 1} k / n} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m n} }\)
\(\ds \) \(=\) \(\ds \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac {m n} e}^{m n} m^{n z - n / 2 + n / 2 - 1 / 2 } } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m n} }\) Sum of Arithmetic Sequence
\(\ds \) \(=\) \(\ds \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^n \paren {\frac m e}^m m^{n z - 1 / 2} n^{1 / 2 - n z} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }\) substituting $m n \mapsto m$
\(\ds \) \(=\) \(\ds \lim_{m \mathop \to \infty} \frac {\paren {\sqrt {2 \pi} }^{n - 1} m! m^{n z - 1} n^{1 / 2 - n z} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }\) Stirling's Formula
\(\ds \) \(=\) \(\ds \lim_{m \mathop \to \infty} \frac {\paren {n z - 1} } {\paren {n z - 1} } \times \frac {\paren {\sqrt {2 \pi} }^{n - 1} m! m^{n z - 1} n^{1 / 2 - n z} } {\paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }\) multiplying top and bottom by $n z - 1$
\(\ds \) \(=\) \(\ds \paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \paren {n z - 1} \lim_{m \mathop \to \infty} \frac {m! m^{n z - 1} } {\paren {n z - 1} \paren {n z} \paren {n z + 1} \cdots \paren {n z - 1 + m} }\) bringing $\paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \paren {n z - 1}$ outside the limit
\(\ds \) \(=\) \(\ds \paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \paren {n z - 1} \map \Gamma {n z - 1}\) Definition of Euler Form of Gamma Function
\(\ds \) \(=\) \(\ds \paren {2 \pi}^{\paren {n - 1} / 2} n^{1 / 2 - n z} \map \Gamma {n z}\) Gamma Difference Equation

$\blacksquare$

Also see


Source of Name

This entry was named for Carl Friedrich Gauss.


Sources

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