False and partial Eisenstein series related to unimodal sequences

TL;DR

This paper explores the properties of false and partial Eisenstein series linked to unimodal sequences, revealing their quasimodular nature, Fourier expansions, and recursive formulas for related generating functions.

Contribution

It introduces a new space of functions related to false and partial Eisenstein series that is closed under differentiation and contains quasimodular forms.

Findings

The space of these objects is closed under differentiation.

Fourier expansions and quasimodular completions are established.

A recursive formula for the Taylor coefficients of the unimodal rank generating function is derived.

Abstract

Motivated by the fact that the classical Jacobi theta function $\vartheta$ is the exponential generating function of the Eisenstein series, we study the exponential Taylor coefficients (in the elliptic variable) of a related natural partial theta function, as well as a false theta function related to the Dedekind eta function. We prove that the space spanned by these objects is closed under differentiation, analogous to the space of quasimodular forms, and that it contains the quasimodular forms themselves. We further provide their Fourier expansions, establish quasimodular completions, and derive a recursive formula for the Taylor coefficients of the logarithm of the unimodal rank generating function, expressed as partition traces of the false and partial objects.