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MathGroup Archive 2001

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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)
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