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The following tabulates the various appearances of the multiplication theorem for finite characteristic; the characteristic zero relations are given further down. In all cases, ''n'' and ''k'' are non-negative integers. For the special case of ''n'' = 2, the theorem is commonly referred to as the '''duplication formula'''.
==Gamma function–Legendre
The duplication formula and the multiplication theorem for the [[gamma function]] are the prototypical examples. The duplication formula for the gamma function is
:<math>
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(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 [[
==Polygamma function, harmonic numbers==
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