nLab Gauss multiplication formula

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Contents

Statement

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

Proposition

For

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

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

the Gamma function Γ()\Gamma(-) satisfies

j=0N1Γ(z+jN)=(2π) 12(N1)N (12Nz)Γ(Nz). \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=2N = 2 this is known as the duplication formula:

Γ(z)Γ(z+1/2)=(2π) 1/22 1/22zΓ(2z). \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):

j=1NΓ(j/2)=G(N/2+1)G(N/2+1/2)G(1/2). \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:

Last revised on June 1, 2021 at 13:32:13. See the history of this page for a list of all contributions to it.