Identities about level 2 Eisenstein series

TL;DR

This paper explores generalized double Eisenstein series, deriving new transformation formulas, expressing them through hyperbolic series, and presenting novel examples to deepen understanding of their properties.

Contribution

It introduces four new transformation formulas for double Eisenstein series and shows their reducibility to hyperbolic series, expanding the theoretical framework.

Findings

Four new transformation formulas for double Eisenstein series

Representation of these series via hyperbolic functions

Presentation of new examples illustrating these properties

Abstract

In this paper we consider certain classes of generalized double Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formula for some double Eisenstein series. We can find that these double Eisenstein series are reducible to infinite series involving hyperbolic functions. Moreover, some interesting new examples are given.