An Elliptic Analogue Of Generalized Cotangent Dirichlet Series And Its Transformation Formulae At Some Integer Arguments

TL;DR

This paper introduces an elliptic analogue of the generalized cotangent Dirichlet series, derives its transformation formulae at specific integers, and connects these results to classical special value evaluations by Berndt.

Contribution

It establishes a new elliptic analogue of the series and provides transformation formulae that extend and relate to previous classical results.

Findings

Derived transformation formulae for the elliptic analogue at integer arguments

Connected the new formulae to classical special value evaluations

Extended Arakawa's transformation formulae to the elliptic setting

Abstract

B.C. Berndt evaluated special values of the cotangent Dirichlet series. T. Arakawa studied a generalization of the series, or generalized cotangent Dirichlet series, and gave its transformation formulae. In this paper, we establish an elliptic analogue of the generalized cotangent Dirichlet series and give its transformation formulae at some integer arguments. As a corollary, we obtain the transformation formulae of the generalized cotangent Dirichlet series at some integer arguments which are the part of Arakawa's transformation formulae. Those transformation formulae give the special values of the cotangent Dirichlet series evaluated by B.C. Berndt.