Multiplication theorem: Difference between revisions

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</math>

for integer ''k'' &ge; 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 [[Dirichlet character]], of the [[Chowla–Selberg formula]].

==Sine function ==

Formally similar duplication formulas hold for the sine function, which are rather simple consequences of the [[List of trigonometric identities|trigonometric identities]]. Here one has the duplication formula

:<math>

\sin(\pi x)\sin\left(\pi\left(x+\frac{1}{2}\right)\right) = \frac{1}{2}\sin(2\pi x)

</math>

and, more generally, for any integer ''k'', one has

:<math>

\sin(\pi x)\sin\left(\pi\left(x+\frac{1}{k}\right)\right) \cdots \sin\left(\pi\left(x+\frac{k-1}{k}\right)\right) = 2^{1-k} \sin(k \pi x)

</math>

==Polygamma function, harmonic numbers==

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==Hurwitz zeta function==

For theThe [[Hurwitz zeta function]] generalizes the polygamma function to non-integer orders, and thus obeys a very similar multiplication theorem:

:<math>k^s\zeta(s)=\sum_{n=1}^k \zeta\left(s,\frac{n}{k}\right),</math>

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:<math>\mathcal{L}_k B_m = \frac{1}{k^m}B_m</math>

It is the fact that the eigenvalues <math>k^{-m}<1</math> that marks this as a [[dissipative system]]: for a non-dissipative [[measure-preserving dynamical system]], the eigenvalues of the transfer operator lie on the [[unit circle]].

One may construct a function obeying the multiplication theorem from any [[totally multiplicative function]]. Let <math>f(n)</math> be totally multiplicative; that is, <math>f(mn)=f(m)f(n)</math> for any integers ''m'', ''n''. Define its [[Fourier series]] as

:<math>g(x)=\sum_{n=1}^\infty f(n) \exp(2\pi inx)</math>