An elliptic analogue of generalized Dedekind–Rademacher sums

Abstract

In this paper we introduce an elliptic analogue of the generalized Dedekind–Rademacher sums which satisfy reciprocity laws. In these sums, Kronecker's double series play a role of elliptic Bernoulli functions. This paper gives an answer to the problem of S. Fukuhara and N. Yui concerning the elliptic Apostol–Dedekind sums. We also mention a relation between the generating function of Kronecker's double series and that of the (Debye) elliptic polylogarithms studied by A. Levin.