On cubic multisections of Eisenstein series

TL;DR

This paper develops a systematic method for generating cubic multisections of Eisenstein series, expressing the resulting series as rational functions of eta functions, and explores their properties through linear transformations.

Contribution

It introduces a new systematic procedure for cubic multisections of Eisenstein series and characterizes the dissection operators using linear transformations.

Findings

Series are rational functions of eta( au) and eta( au)

Dissection operators mirror properties of quintic operators

Provides a general framework for cubic dissection formulas

Abstract

A systematic procedure for generating cubic multisections of Eisenstein series is given. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo three. We prove that the resulting series are rational functions of \eta(\tau) and \eta(3\tau), where \eta is the Dedekind eta function. A more general treatment of cubic dissection formulas is given by describing the dissection operators in terms of linear transformations. These operators exhibit properties that mirror those of similarly defined quintic operators.