Multiplication theorem: Difference between revisions

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{{short description|Identity obeyed by many special functions related to the gamma function}}

{{This|the identity obeyed by special functions related to the gamma function|the multiplication rule in probability theory|Independence (probability theory)}}

{{More refs|date=February 2021}}

In [[mathematics]], the '''multiplication theorem''' is a certain type of identity obeyed by many [[special function]]s related to the [[gamma function]]. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises.

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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-Legendrefunction–Legendre functionformula ==

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^{\frac{1/-2kz}{2} - kz} \; \Gamma(kz) \,\!

</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 [[trivialDirichlet character]], of the [[Chowla–Selberg formula]].

==Sine function ==

This implies a variation of [[Stirling's approximation]] that is exact --- a '''Stirling identity''' ---

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>

\expsin(\pi x)\sin\left(\int_z^{zpi\left(x+\frac{1}{2}\log\Gamma(xright)\,dx\right) = \sqrtfrac{1}{2}\sin(2\pi} z^ze^{-z}.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>

The left side is the '''geometric mean''' of <math>\Gamma</math> over the line segment <math>[z,z+1]</math>.

==Polygamma function, harmonic numbers==

The [[polygamma function]] is the [[logarithmic derivative]] of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative:

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:<math>k\left[\psi(kz)-\log(k)\right] = \sum_{n=0}^{k-1}

\psi\left(z+\frac{n}{k}\right).</math>

The polygamma identities can be used to obtain a multiplication theorem for [[harmonic number]]s.

==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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where Li_''s''(''z'') is the [[polylogarithm]]. It obeys the duplication formula

:<math>2^{1-s} F(s;q) = F\left(s,\frac{q}{2}\right)

+ F\left(s,\frac{q+1}{2}\right).</math>

As such, it is an eigenvector of the [[Bernoulli operator]] with eigenvalue 2^{1&minus;''s''}. The multiplication theorem is

:<math>k^{1-s} F(s;kq) = \sum_{n=0}^{k-1} F\left(s,q+\frac{n}{k}\right).</math>

The periodic zeta function occurs in the reflection formula for the Hurwitz zeta function, which is why the relation that it obeys, and the Hurwitz zeta relation, differ by the interchange of&nbsp;''s''&nbsp;&rarr;&nbsp;1&minus;''s''.

The [[Bernoulli polynomials]] may be obtained as a limiting case of the periodic zeta function, taking ''s'' to be an integer, and thus the multiplication theorem there can be derived from the above. Similarly, substituting&nbsp;''q''&nbsp;=&nbsp;log&nbsp;''z'' leads to the multiplication theorem for the polylogarithm.

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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)\ </math>

and thus resembles that for the polylogarithm, but twisted by&nbsp;''i''.

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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>

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:<math>\frac{1}{k}\sum_{n=0}^{k-1}g\left(\frac{x+n}{k}\right)=f(k)g(x)</math>

That is, ''g''(''x'') is an eigenfunction of Bernoulli transfer operator, with eigenvalue ''f''(''k''). The multiplication theorem for the Bernoulli polynomials then follows as a special case of the multiplicative function <math>f(n)=n^{-s}</math>. The [[Dirichlet character]]s are fully multiplicative, and thus can be readily used to obtain additional identities of this form.

==Characteristic zero==