 
Summary: PolyGamma Functions of Negative Order
VICTOR S. ADAMCHIK
January 22, 1998
Abstract
Liouville's fractional integration is used to define polygamma func
tions / (n) (z) for negative integer n. It's shown that such / (n) (z) can
be represented in a closed form by means of the first derivatives of
the Hurwitz Zeta function. Relations to the Barnes Gfunction and
generalized Glaisher's constants are also discussed.
1 Introduction
The idea to define the polygamma function / (š) (z) for every complex š via
Liouville's fractional integration operator is quite natural and was around
for a while (see Ross (1974) and Grossman (1976)). However, for arbitrary
negative integer š the closed form of / (š) (z) was not developed yet the
only two particular cases š = \Gamma2 and š = \Gamma3 have been studied (see Gosper
(1997)). It is the purpose of this note is to consider
/ (\Gamman) (z) = 1
(n \Gamma 2)!
Z z
0
