Multiplication theorem
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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
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 formula
The duplication formula and the multiplication theorem for the gamma function are the prototypical examples. The duplication formula for the gamma function is
It is also called the Legendre duplication formula[1] or Legendre relation, in honor of Adrien-Marie Legendre. The multiplication theorem is
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 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 trigonometric identities. Here one has the duplication formula
and, more generally, for any integer k, one has
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:
for , and, for , one has the digamma function:
The polygamma identities can be used to obtain a multiplication theorem for harmonic numbers.
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
where Lis(z) is the polylogarithm. It obeys the duplication formula
As such, it is an eigenvector of the
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.
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:
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
and
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
Perhaps not surprisingly, the
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
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 . The Dirichlet characters are fully multiplicative, and thus can be readily used to obtain additional identities of this form.
Characteristic zero
The multiplication theorem over a field of
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.