Some three-term and mixed three-term relations for Hardy sums were given by Goldberg . His proofs are based on Bernd's transformation formulae for the logarithms of the classical Theat-functions. Pettet and Sitaramachandararo  proved elementary proofs for all of Goldberg's results and also proved some three-term relations of Dedekind sums. In this paper, some new theorems on three-term relations for hardy sums were found by applying derivative operator to three-term polynomial relation. Furthermore, proofs of the reciprocity relations for Hardy sums are presented in a more concise way from the original proofs of Berndt [2, 3, 4] and Goldberg .
ŞİMŞEK, Y. (1998) "Theorems on Three-Term Relations for Hardy Sum," Turkish Journal of Mathematics: Vol. 22: No. 2, Article 3. Available at: https://journals.tubitak.gov.tr/math/vol22/iss2/3