Summary: PolyGamma Functions of Negative Order
VICTOR S. ADAMCHIK
January 22, 1998
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.
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)!