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False and partial Eisenstein-type series related to unimodal sequences
Kathrin Bringmann1, Badri Vishal Pandey1, Jan-Willem van Ittersum1,2
1Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany.
Research in the Mathematical Sciences
|May 11, 2026
Summary
This study explores partial and false theta functions, revealing their connection to quasimodular forms. Their properties, including Fourier expansions and recursive formulas, are established for partition theory applications.
Area of Science:
- Number Theory
- Combinatorics
- Mathematical Physics
Background:
- The Jacobi theta function serves as an exponential generating function for Eisenstein series.
- Quasimodular forms are a generalization of modular forms with important applications.
Purpose of the Study:
- To investigate the properties of partial and false theta functions related to the Dedekind eta function.
- To demonstrate that these functions span a space closed under differentiation, analogous to quasimodular forms.
- To derive a recursive formula for coefficients of the unimodal rank generating function.
Main Methods:
- Analysis of exponential Taylor coefficients in the elliptic variable.
- Fourier expansion of theta functions.
- Establishing quasimodular completions.
- Deriving recursive formulas for partition traces.
Main Results:
- The space spanned by the studied theta functions is closed under differentiation.
- This space contains quasimodular forms.
- Fourier expansions and quasimodular completions are provided.
- A recursive formula for the logarithm of the unimodal rank generating function is derived.
Conclusions:
- Partial and false theta functions exhibit quasimodular properties.
- These findings offer new insights into the relationship between theta functions and quasimodular forms.
- The derived recursive formula has implications for partition theory and related combinatorial problems.
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