Digamma Function - Gaussian Sum

Gaussian Sum

The digamma has a Gaussian sum of the form

\frac{-1}{\pi k} \sum_{n=1}^k
\sin \left( \frac{2\pi nm}{k}\right) \psi \left(\frac{n}{k}\right) =
\zeta\left(0,\frac{m}{k}\right) = -B_1 \left(\frac{m}{k}\right) =
\frac{1}{2} - \frac{m}{k}

for integers . Here, ΞΆ(s,q) is the Hurwitz zeta function and is a Bernoulli polynomial. A special case of the multiplication theorem is

\sum_{n=1}^k \psi \left(\frac{n}{k}\right) =-k(\gamma+\log k),

and a neat generalization of this is

where q must be a natural number, but 1-qa not.

Read more about this topic:  Digamma Function

Famous quotes containing the word sum:

    Genius is no more than childhood recaptured at will, childhood equipped now with man’s physical means to express itself, and with the analytical mind that enables it to bring order into the sum of experience, involuntarily amassed.
    Charles Baudelaire (1821–1867)