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The duplication formula and the multiplication theorem for the [[gamma function]] are the prototypical examples. The duplication formula for the gamma function is
:<math>
\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z).
</math>
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\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)
</math>
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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The duplication formula for [[Kummer's function]] is
:<math>2^{1-n}\Lambda_n(-z^2) = \Lambda_n(z)+\Lambda_n(-z)
and thus resembles that for the polylogarithm, but twisted by ''i''.
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