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The Quarterly Journal of Mathematics Advance Access published online on March 17, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap004
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

AN INTEGRAL REPRESENTATION OF MULTIPLE HURWITZ–LERCH ZETA FUNCTIONS AND GENERALIZED MULTIPLE BERNOULLI NUMBERS

Yasushi Komori{dagger}

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464–8602, Japan

{dagger} Corresponding author. E-mail: komori{at}math.nagoya-u.ac.jp

Received 1 August 2008; revised 10 December 2008
  Abstract

A surface integral representation of a multiple generalization of the Hurwitz–Lerch zeta function is given, which is a direct analogue of the well-known contour integral representation of the Riemann zeta function of Hankel's type. From this integral representation, we derive a detailed description of the set of its possible singularities. In addition, we present two formulae for special values of the zeta function at non-positive integers in terms of generalizations of Bernoulli numbers. These results are refinements of previously known ones.


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