In the first case, a finite number of terms are added or multiplied to give the relation.
The infinite sums are much more common, and follow from characteristic zero relations on the hypergeometric series.
For the special case of n = 2, the theorem is commonly referred to as the duplication formula.
The duplication formula and the multiplication theorem for the gamma function are the prototypical examples.
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.
Formally similar duplication formulas hold for the sine function, which are rather simple consequences of the trigonometric identities.
, one has the digamma function: The polygamma identities can be used to obtain a multiplication theorem for harmonic numbers.
This is a special case of and Multiplication formulas for the non-principal characters may be given in the form of Dirichlet L-functions.
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.
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.
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).
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 Dirichlet characters are fully multiplicative, and thus can be readily used to obtain additional identities of this form.
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.