gaussmult

Gauss's multiplication formula.

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.

The multiplication theorem is:

G(a)*G(a+1/b)*G(a+2/b)*...*G(a + b-1/b) = prod_[k=0 to b-1] G(a + k/b)

= (2*pi)^(b/2)*b^(1/2-ab)*G(ab)

where G is the gamma function.

For integer k >= 1, and is sometimes called Gauss's multiplication formula, in honour of Carl Friedrich Gauss.