WordNet

a multiplicative increase; "repeated copying leads to a multiplication of errors"; "this multiplication of cells is a natural correlate of growth"
an arithmetic operation that is the inverse of division; the product of two numbers is computed; "the multiplication of four by three gives twelve"; "four times three equals twelve" (同)times
a proposition deducible from basic postulates
an idea accepted as a demonstrable truth

PrepTutorEJDIC

〈U〉(…の数・量の)増加,(動植物の)増殖,繁殖《+『of』+『名』》 / 〈U〉〈C〉乗法,掛け算
定理 / (一般に)自明の原理,定理

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2015/02/18 19:20:46」(JST)

wiki en

In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions 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.

Finite characteristic

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 a p-adic relation over a finite field. For example, the multiplication theorem for the gamma function follows from the Chowla–Selberg formula, which follows from the theory of complex multiplication. The infinite sums are much more common, and follow from characteristic 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 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 function

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

\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z). \,\!

It is also called the Legendre duplication formula^[1] or Legendre relation, in honor of Adrien-Marie Legendre. The multiplication theorem is

\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) \,\!

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.

Polygamma function

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

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

for , and, for , one has the digamma function:

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

Hurwitz zeta function

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

where is the Riemann zeta function. This is a special case of

and

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

Periodic zeta function

The periodic zeta function^[2] is sometimes defined as

F(s;q) = \sum_{m=1}^\infty \frac {e^{2\pi imq}}{m^s} =\operatorname{Li}_s\left(e^{2\pi i q} \right)

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

2^{-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 the Bernoulli operator with eigenvalue 2^−s. The multiplication theorem is

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 → −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 q = log z leads to the multiplication theorem for the polylogarithm.

Polylogarithm

The duplication formula takes the form

The general multiplication formula is in the form of a Gauss sum or discrete Fourier transform:

k^{1-s} \operatorname{Li}_s(z^k) = \sum_{n=0}^{k-1}\operatorname{Li}_s\left(ze^{i2\pi n/k}\right).

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

Kummer's function

The duplication formula for Kummer's function is

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

Bernoulli polynomials

For the Bernoulli polynomials, the multiplication theorems were given by Joseph Ludwig Raabe in 1851:

and for the Euler polynomials,

k^{-m} E_m(kx)= \sum_{n=0}^{k-1} (-1)^n E_m \left(x+\frac{n}{k}\right) \quad \mbox{ for } k=1,3,\dots

and

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) \quad \mbox{ 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.

Bernoulli map

The Bernoulli map is a certain simple model of a dissipative dynamical system, describing the effect of a shift operator on an infinite string of coin-flips (the Cantor set). The Bernoulli map is a one-sided version of the closely related Baker's map. The Bernoulli map generalizes to a k-adic version, which acts on infinite strings of k symbols: this is the Bernoulli scheme. The transfer operator corresponding to the shift operator on the Bernoulli scheme is given by

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

It is the fact that the eigenvalues 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 be totally multiplicative; that is, for any integers m, n. Define its Fourier series as

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

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 .

Characteristic zero

The multiplication theorem over a field of characteristic zero does not close after a finite number of terms, but requires an infinite series to be expressed. Examples include that for the Bessel function :

\lambda^{-\nu} J_\nu (\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{(1-\lambda^2)z}{2}\right)^n J_{\nu+n}(z),

where and may be taken as arbitrary complex numbers. Such characteristic-zero identities follow generally from one of many possible identities on the hypergeometric series.

Notes

^ Weisstein, Eric W., "Legendre Duplication Formula", MathWorld.
^ Apostol, Introduction to analytic number theory, Springer

References

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.

UpToDate Contents

全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.

1. レジオネラ感染の疫学および病因 epidemiology and pathogenesis of legionella infection
2. 異常ヘモグロビン症におけるマラリアに対する防御 protection against malaria in the hemoglobinopathies
3. 炭疽菌の微生物学、病因、および疫学 microbiology pathogenesis and epidemiology of anthrax
4. 痘瘡の疫学、病因、および臨床症状 the epidemiology pathogenesis and clinical manifestations of smallpox
5. コロモジラミ症 pediculosis corporis

English Journal

Simulations of operation dynamics of different type GaN particle sensors.

Gaubas E1, Ceponis T2, Kalesinskas V3, Pavlov J4, Vysniauskas J5.
Sensors (Basel, Switzerland).Sensors (Basel).2015 Mar 5;15(3):5429-73. doi: 10.3390/s150305429.
The operation dynamics of the capacitor-type and PIN diode type detectors based on GaN have been simulated using the dynamic and drift-diffusion models. The drift-diffusion current simulations have been implemented by employing the software package Synopsys TCAD Sentaurus. The monopolar and bipolar
PMID 25751080

A stochastic model for electron multiplication charge-coupled devices--from theory to practice.

Hirsch M1, Wareham RJ, Martin-Fernandez ML, Hobson MP, Rolfe DJ.
PloS one.PLoS One.2013;8(1):e53671. doi: 10.1371/journal.pone.0053671. Epub 2013 Jan 31.
Electron multiplication charge-coupled devices (EMCCD) are widely used for photon counting experiments and measurements of low intensity light sources, and are extensively employed in biological fluorescence imaging applications. These devices have a complex statistical behaviour that is often not f
PMID 23382848

A fast algorithm for computing binomial coefficients modulo powers of two.

Andreica MI1.
TheScientificWorldJournal.ScientificWorldJournal.2013 Nov 6;2013:751358. doi: 10.1155/2013/751358. eCollection 2013.
I present a new algorithm for computing binomial coefficients modulo 2N. The proposed method has an O(N3·Multiplication(N)+N4) preprocessing time, after which a binomial coefficient C(P, Q) with 0≤Q≤P≤2N-1 can be computed modulo 2N in O(N2·log(N)·Multiplication(N)) time. Multiplication(N) d
PMID 24348186

Japanese Journal

``Average-Value Paradox of the Lognormal Distribution

YAMAMOTO Ken,WAKITA Jun-ichi
J Phys Soc Jpn 82(11), 113001-113001-3, 2013-11-15
… The authors consider a simple stochastic process given by multiplication of uniform random variables. … The central limit theorem ensures that a lognormal distribution is asymptotically a good approximation to the actual distribution. …
NAID 40019871145

エントロピーと非加法性(一般セッション,機械学習と視覚情報処理の接点,及び,社会テーマ:ハイリスク作業支援)

田中勝
電子情報通信学会技術研究報告. PRMU, パターン認識・メディア理解 113(196), 39-46, 2013-08-26
本報告では,くり込まれたγ一対数尤度との縮約を用いてエントロピーが与えられ,正規化することで共形エントロピーが定義される.このエントロピーとツァリスエントロピーは簡単な関係で結びついている.また,ここで定義されたエントロピーは,τ-対数関数の積に関する恒等式が2種類存在するために,それに応じて2種類の非加法性をもつことができる.どちらが適切かは実験によって決められるべきであるが,共形性を重視すれば …
NAID 110009824940

エントロピーと非加法性(一般セッション,機械学習と視覚情報処理の接点,及び,社会テーマ:ハイリスク作業支援)

田中勝
電子情報通信学会技術研究報告. IBISML, 情報論的学習理論と機械学習 113(197), 39-46, 2013-08-26
本報告では,くり込まれたγ-対数尤度との縮約を用いてエントロピーが与えられ,正規化することで共形エントロピーが定義される.このエントロピーとツァリスエントロピーは簡単な関係で結びついている.また,ここで定義されたエントロピーは,τ-対数関数の積に関する恒等式が2種類存在するために,それに応じて2種類の非加法性をもつことができる.どちらが適切かは実験によって決められるべきであるが,共形性を重視すれば …
NAID 110009784533

「乗法定理」

　　[★]

英: multiplication theorem

[1] Weisstein, Eric W., "Legendre Duplication Formula", MathWorld.

[2] Apostol, Introduction to analytic number theory, Springer

匿名

検索

案内

案内

multiplication theorem