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False and partial Eisenstein series related to unimodal sequences

Kathrin Bringmann, Badri Vishal Pandey, Jan-Willem van Ittersum

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.

False and partial Eisenstein series related to unimodal sequences

Abstract

Motivated by the fact that the classical Jacobi theta function 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.
Paper Structure (18 sections, 21 theorems, 123 equations)

This paper contains 18 sections, 21 theorems, 123 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminaries
  3. Combinatorial lemmas
  4. The Dedekind eta function and (false) theta functions
  5. Quasimodular forms
  6. Proofs of Theorems \ref{['thm:comp-of-g_k']} and \ref{['thm:comp-of-h_k']}
  7. Modularity
  8. Modular completions of $g_k$ and $h_k$
  9. Proofs of Theorems \ref{['thm:mathbbT-algebra']} and \ref{['thm:h-n-diff']}
  10. Proof of Theorem \ref{['thm:mathbbT-algebra']}
  11. Proof of Theorem \ref{['thm:h-n-diff']}
  12. Proofs of Theorems \ref{['thm:Fourier-exp-of-g_k']} and \ref{['thm:Fourier-exp-of-h_k']}
  13. Constant terms
  14. Proof of Theorem \ref{['thm:Fourier-exp-of-g_k']}
  15. Proof of Theorem \ref{['thm:Fourier-exp-of-h_k']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

There exist functions $\widehat{g}_k(\tau,w)$ such that the following holds:

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark
  • Corollary 1.8
  • Lemma 2.1
  • ...and 23 more