Non-Integer HarmonicNumbers?

*To*: mathgroup at smc.vnet.net*Subject*: [mg29929] Non-Integer HarmonicNumbers?*From*: Dirk Reiss <reiss at physnet.uni-hamburg.de>*Date*: Thu, 19 Jul 2001 03:56:55 -0400 (EDT)*Organization*: University of Hamburg -- Germany*Sender*: owner-wri-mathgroup at wolfram.com

Hi I encountered the combinatorial function HarmonicNumber, which according to the documentation is defined as a FINITE sum HarmonicNumber[N,x]=Sum[n^(-x),{n,1,N}]. (It therefore is something one could call an incomplete Zeta function.) Apart from the definition the documenation does not give much help, e.g. relations to other special funtions. However, I think there must be some analytic continuation, since Mathematica is able to compute the the HarmonicNumber for non-integer values for N, which does not make sense from the sum definition. Does anyone know about this, i.e. how does Mathematica compute this Numbers? Thank you in advance Dirk ---------------------------------------------------------------------- Dr. Dirk Reiss Institut fuer Laser-Physik Tel: (040) 42838-2908 Universitaet Hamburg Fax: (040) 42838-6571 Jungiusstrasse 9 email: reiss at physnet.uni-hamburg.de D-20359 Hamburg GERMANY (http://www.physnet.uni-hamburg.de/ilp/english/reiss.html) ----------------------------------------------------------------------