Multiplication Theorem - Gamma Function-Legendre Function

Gamma Function-Legendre Function

The duplication formula and the multiplication theorem for the gamma function are the prototypical examples. The duplication formula for the gamma function is

\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z). \,\!

It is also called the Legendre duplication formula or Legendre relation, in honor of Adrien-Marie Legendre. The multiplication theorem is

\Gamma(z) \; \Gamma\left(z + \frac{1}{k}\right) \; \Gamma\left(z + \frac{2}{k}\right) \cdots \Gamma\left(z + \frac{k-1}{k}\right) = (2 \pi)^{ \frac{k-1}{2}} \; k^{1/2 - kz} \; \Gamma(kz) \,\!

for integer k ≥ 1, and is sometimes called Gauss's multiplication formula, in honour of Carl Friedrich Gauss. The multiplication theorem for the gamma functions can be understood to be a special case, for the trivial character, of the Chowla–Selberg formula.

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