Generalized Hardy-Berndt sums
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In this paper, explicit generalizations of Hardy-Berndt sums, which are arising in the transformation formula of the logarithms of the classical theta functions, are de¯ned. The connection between generalized Dedekind sums is also given, and some ¯nite and in¯nite series representations are obtained in terms of trigonometric functions and Hurwitz's zeta function. By using contour integration, the reciprocity laws and some relations for these generalized sums are given as well.
Abstract
In this paper, explicit generalizations of Hardy-Berndt sums, which are arising in the transformation formula of the logarithms of the classical theta functions, are de¯ned. The connection between generalized Dedekind sums is also given, and some ¯nite and in¯nite series representations are obtained in terms of trigonometric functions and Hurwitz's zeta function. By using contour integration, the reciprocity laws and some relations for these generalized sums are given as well.