Hurwitz Zeta Function - Special Cases and Generalizations

Special Cases and Generalizations

The Hurwitz zeta function with non-negative integer m is related to the polygamma function:

For negative integer −n the values are related to the Bernoulli polynomials:

The Barnes zeta function generalizes the Hurwitz zeta function.

The Lerch transcendent generalizes the Hurwitz zeta:

\Phi(z, s, q) = \sum_{k=0}^\infty \frac { z^k} {(k+q)^s}

and thus

Hypergeometric function

where

Meijer G-function

Read more about this topic:  Hurwitz Zeta Function

Famous quotes containing the words special and/or cases:

    “The rebellion is against time pollution, the feeling that the essence of what makes life worth living—the small moments, the special family getaways, the cookies in the oven, the weekend drives, the long dreamlike summers Mso much of this has been taken from us, or we have given it up. For what? Hitachi stereos? Club Med? Company cars? Racquetball? For fifteen-hour days and lousy day care?”
    —Richard Louv (20th century)

    “... and the next summer she died in childbirth.
    That’s all. Of course, there may be some sort of sequel but it is not known to me. In such cases instead of getting bogged down in guesswork, I repeat the words of the merry king in my favorite fairy tale: Which arrow flies for ever? The arrow that has hit its mark.”
    —Vladimir Nabokov (1899–1977)