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Inmathematics, themultiplication theorem is a certain type of identity obeyed by manyspecial functions related to thegamma 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.

The multiplication theorem takes two common forms. In the first case, a finite number of terms are added or multiplied to give the relation. In the second case, an infinite number of terms are added or multiplied. The finite form typically occurs only for the gamma and related functions, for which the identity follows from ap-adic relation over afinite field. For example, the multiplication theorem for the gamma function follows from theChowla–Selberg formula, which follows from the theory ofcomplex multiplication. The infinite sums are much more common, and follow fromcharacteristic zero relations on the hypergeometric series.

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 andk are non-negative integers. For the special case ofn = 2, the theorem is commonly referred to as theduplication formula.

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

${\displaystyle \mathrm{\Gamma }(z)\mathrm{\Gamma }\left(z+\frac{1}{2}\right)=2^{1-2z}\sqrt{\pi }\mathrm{\Gamma }(2z).}$

It is also called theLegendre duplication formula^[1] orLegendre relation, in honor ofAdrien-Marie Legendre. The multiplication theorem is

${\displaystyle \mathrm{\Gamma }(z)\mathrm{\Gamma }\left(z+\frac{1}{k}\right)\mathrm{\Gamma }\left(z+\frac{2}{k}\right)\cdots \mathrm{\Gamma }\left(z+\frac{k-1}{k}\right)=(2\pi {)}^{\frac{k-1}{2}}{k}^{\frac{1-2kz}{2}}\mathrm{\Gamma }(kz)}$

for integerk ≥ 1, and is sometimes calledGauss's multiplication formula, in honour ofCarl Friedrich Gauss. The multiplication theorem for the gamma functions can be understood to be a special case, for the trivialDirichlet character, of theChowla–Selberg formula.

Formally similar duplication formulas hold for the sine function, which are rather simple consequences of thetrigonometric identities. Here one has the duplication formula

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

and, more generally, for any integerk, one has

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

Thepolygamma function is thelogarithmic derivative of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative:

${\displaystyle k^m{\psi }^{(m-1)}(kz)=\sum _{n=0}^{k-1}{\psi }^{(m-1)}\left(z+\frac{n}{k}\right)}$

for${\displaystyle m>1}$, and, for${\displaystyle m=1}$, one has thedigamma function:

${\displaystyle k\left[\psi (kz)-\log (k)\right]=\sum _{n=0}^{k-1}\psi \left(z+\frac{n}{k}\right).}$

The polygamma identities can be used to obtain a multiplication theorem forharmonic numbers.

TheHurwitz zeta function generalizes the polygamma function to non-integer orders, and thus obeys a very similar multiplication theorem:

${\displaystyle k^s\zeta (s)=\sum _{n=1}^k\zeta \left(s,\frac{n}{k}\right),}$

where${\displaystyle \zeta (s)}$ is theRiemann zeta function. This is a special case of

${\displaystyle k^s\zeta (s,kz)=\sum _{n=0}^{k-1}\zeta \left(s,z+\frac{n}{k}\right)}$

and

${\displaystyle \zeta (s,kz)=\sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{s+n-1}{n})(1-k{)}^n{z}^n\zeta (s+n,z).}$

Multiplication formulas for the non-principal characters may be given in the form ofDirichlet L-functions.

Theperiodic zeta function^[2] is sometimes defined as

${\displaystyle F(s;q)=\sum _{m=1}^{\mathrm{\infty }}\frac{e^{2\pi imq}}{m^s}={\mathrm{Li}}_s\left(e^{2\pi iq}\right)}$

where Li_s(z) is thepolylogarithm. It obeys the duplication formula

${\displaystyle 2^{1-s}F(s;q)=F\left(s,\frac{q}{2}\right)+F\left(s,\frac{q+1}{2}\right).}$

As such, it is an eigenvector of theBernoulli operator with eigenvalue 2^1−s. The multiplication theorem is

${\displaystyle k^{1-s}F(s;kq)=\sum _{n=0}^{k-1}F\left(s,q+\frac{n}{k}\right).}$

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 s → 1−s.

TheBernoulli polynomials may be obtained as a limiting case of the periodic zeta function, takings to be an integer, and thus the multiplication theorem there can be derived from the above. Similarly, substituting q = log z leads to the multiplication theorem for the polylogarithm.

The duplication formula takes the form

${\displaystyle 2^{1-s}{\mathrm{Li}}_s(z^2)={\mathrm{Li}}_s(z)+{\mathrm{Li}}_s(-z).}$

The general multiplication formula is in the form of aGauss sum ordiscrete Fourier transform:

${\displaystyle k^{1-s}{\mathrm{Li}}_s(z^k)=\sum _{n=0}^{k-1}{\mathrm{Li}}_s\left(z{e}^{i2\pi n/k}\right).}$

These identities follow from that on the periodic zeta function, taking z = log q.

The duplication formula forKummer's function is

${\displaystyle 2^{1-n}{\mathrm{\Lambda }}_n(-z^2)={\mathrm{\Lambda }}_n(z)+{\mathrm{\Lambda }}_n(-z)}$

and thus resembles that for the polylogarithm, but twisted by i.

For theBernoulli polynomials, the multiplication theorems were given byJoseph Ludwig Raabe in 1851:

${\displaystyle k^{1-m}{B}_m(kx)=\sum _{n=0}^{k-1}{B}_m\left(x+\frac{n}{k}\right)}$

and for theEuler polynomials,

${\displaystyle k^{-m}{E}_m(kx)=\sum _{n=0}^{k-1}(-1{)}^n{E}_m\left(x+\frac{n}{k}\right)\text{ for }k=1,3,\dots }$

and

${\displaystyle k^{-m}{E}_m(kx)=\frac{-2}{m+1}\sum _{n=0}^{k-1}(-1{)}^n{B}_{m+1}\left(x+\frac{n}{k}\right)\text{ for }k=2,4,\dots .}$

The Bernoulli polynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there.

TheBernoulli map is a certain simple model of adissipativedynamical system, describing the effect of ashift operator on an infinite string of coin-flips (theCantor set). The Bernoulli map is a one-sided version of the closely relatedBaker's map. The Bernoulli map generalizes to ak-adic version, which acts on infinite strings ofk symbols: this is theBernoulli scheme. Thetransfer operator${\displaystyle {\mathcal{L}}_k}$ corresponding to the shift operator on the Bernoulli scheme is given by

${\displaystyle [{\mathcal{L}}_{k}f](x)=\frac{1}{k}\sum _{n=0}^{k-1}f\left(\frac{x+n}{k}\right)}$

Perhaps not surprisingly, theeigenvectors of this operator are given by the Bernoulli polynomials. That is, one has that

${\displaystyle {\mathcal{L}}_k{B}_m=\frac{1}{k^m}{B}_m}$

It is the fact that the eigenvalues that marks this as adissipative system: for a non-dissipativemeasure-preserving dynamical system, the eigenvalues of the transfer operator lie on theunit circle.

One may construct a function obeying the multiplication theorem from anytotally multiplicative function. Let${\displaystyle f(n)}$ be totally multiplicative; that is,${\displaystyle f(mn)=f(m)f(n)}$ for any integersm,n. Define itsFourier series as

${\displaystyle g(x)=\sum _{n=1}^{\mathrm{\infty }}f(n)\exp (2\pi inx)}$

Assuming that the sum converges, so thatg(x) exists, one then has that it obeys the multiplication theorem; that is, that

${\displaystyle \frac{1}{k}\sum _{n=0}^{k-1}g\left(\frac{x+n}{k}\right)=f(k)g(x)}$

That is,g(x) is an eigenfunction of Bernoulli transfer operator, with eigenvaluef(k). The multiplication theorem for the Bernoulli polynomials then follows as a special case of the multiplicative function${\displaystyle f(n)=n^{-s}}$. TheDirichlet characters are fully multiplicative, and thus can be readily used to obtain additional identities of this form.

The multiplication theorem over a field ofcharacteristic zero does not close after a finite number of terms, but requires aninfinite series to be expressed. Examples include that for theBessel function${\displaystyle J_{\nu }(z)}$:

${\displaystyle {\lambda }^{-\nu }{J}_{\nu }(\lambda z)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\left(\frac{(1-{\lambda }^2)z}{2}\right)}^n{J}_{\nu +n}(z),}$

where${\displaystyle \lambda }$ and${\displaystyle \nu }$ may be taken as arbitrary complex numbers. Such characteristic-zero identities follow generally from one of many possible identities on the hypergeometric series.

• Milton Abramowitz and Irene A. Stegun, eds.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (1972) Dover, New York.(Multiplication theorems are individually listed chapter by chapter)
• C. Truesdell, "On the Addition and Multiplication Theorems for the Special Functions",Proceedings of the National Academy of Sciences, Mathematics, (1950) pp. 752–757.

• Special functions
• Zeta and L-functions
• Gamma and related functions
• Mathematical theorems

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