An elliptic analogue of generalized Dedekind-Rademacher sums (Q958673)

The author introduces an elliptic analogue of the generalized Dedekind-Rademacher sums. In these sums, Kronecker's double series play a role of elliptic Bernoulli functions. This paper gives an answer to the problem of \textit{S. Fukuhara} and \textit{N. Yui} [Trans. Am. Math. Soc. 356, No. 10, 4237--4254 (2004; Zbl 1055.11028)] concerning the elliptic Apostol-Dedekind sums. The main aim of this paper is to establish the reciprocity laws for the elliptic Dedekind-Rademacher sums (Theorem 3.1). As a corollary, the author reproduces the reciprocity laws for the classical generalized Dedekind-Rademacher sums (Theorem 3.2). He also mentions a relation between the generating function of Kronecker's double series and that of the (Debye) elliptic polylogarithms studied by \textit{A. Levin} [Compos. Math. 106, No. 3, 267--282 (1997; Zbl 0905.11028)].