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 ApostolDedekind 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.

The Jacobi form D_L, theta functions, and Eisenstein series

“…Let L be a complex lattice and note by πA = a(L) the area of the parallelogram period of L , and set E L (s, t) := 1 2πiA (st − st) for any complex s, t. For the lattice L the analogue of the expressions (1.6) is given by [4,6,5]. In Section 5, for | x |< d(L, z) these quantities satisfy…”

“…Those elliptic analogue are considered by K. Katayama [4] to prove Von Staudt congruences, also T. Machide in [5] obtained Dedekind reciprocity laws and in [6] proved recurrences formulas of order ≥ 2.…”

Sums of products of Kronecker's double series

“…Ivashkevich et al [6, Section 3.1] and K. Katayama [8] noted that these series can be considered as an elliptic generalization of the classical Bernoulli functions. The author [12] mentioned a relation between the generating function of Kronecker's double series and that of the (Debye) elliptic polylogarithms studied by A. Levin [11] in order to enforce the validity of their elliptic generalization.…”

Generating functions of even Dedekind symbols with polynomial reciprocity laws

“…The function B m (x, y; τ ) is called Kronecker's double series (or the elliptic Bernoulli function by Machide [6]). Note that B m (x, y; τ ) can be expressed as…”

“…Theorem 2.1 (Machide [6]). Let a, a , b, b , c, c be positive integers, and x, y, z be real numbers.…”