nLab Gauss multiplication formula

Statement

The following is known as the Gauss multiplication formula or the Legendre relation:

Proposition

For

• a positive integer $N \in \mathbb{N}_+$,

• any $z \in \mathbb{R} \setminus \{0, -1/N, -2/N, \cdots\}$,

the Gamma function $\Gamma(-)$ satisfies

$$\underoverset {j = 0} {N-1} {\prod} \Gamma \left( z + \tfrac{j}{N} \right) \;=\; (2 \pi)^{ \tfrac{1}{2}(N-1) } \cdot N^{ (\tfrac{1}{2} - N z) } \cdot \Gamma( N z ) \,.$$

Example

In the special case of $N = 2$ this is known as the duplication formula:

$$\Gamma(z) \cdot \Gamma \big( z + 1/2 \big) \;=\; (2\pi)^{1/2} \cdot 2^{1/2 - 2z} \cdot \Gamma(2 z) \,.$$

Remark

A related expression gives a product of Barnes G-functions (see there):

$$\underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \frac { G(N/2+ 1) \cdot G(N/2 + 1/2) } { G(1/2) } \,.$$

References

• J. Sándor and L. Tóth, A remark on the gamma function, Elem. Math. 44 (3), pp. 73–76 (1989) (dml:141455)

See also:

• Wikipedia, Multiplication theorem

• ProofWiki, Gauss Multiplication Formula

• DLMF, Gamma function §5.5(iii) Multiplication

• Wolfram MathWorld, Gauss multiplication formula

• Wolfram MathWorld, Legendre Duplication formula