Reciprocity Formulae for Generalized Dedekind Sums

Dedekind sums are finite arithmetic sums that appear in diverse fields, like number theory and computer science.  For example, they can be used in random number generation.  In 1995, Hall, Wilson, and Zagier introduced a more general version of Dedekind sums involving periodized Bernoulli polynomials.  Their sum became known as the Hall-Wilson-Zagier (HWZ) sum.  Hall, Wilson, and Zagier proved a reciprocity theorem for the generating function of the HWZ sum that implied known classical reciprocity theorems on Dedekind sums proven by Rademacher and Dedekind himself.  In 1993,  Pommersheim proved a new reciprocity theorem for the Dedekind sum that later was shown by  Girstmair to follow from Rademacher's and Dedekind's work.  We investigate analogues of the HWZ theorem with regard to Pommersheim's reciprocity theorem.  Our goal is to apply Girstmair's methodology to the generating function of the HWZ sum in order to obtain the generating function reciprocity corresponding to Pommersheim's theorem.