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A Bijective Proof of a False Theta Function Identity from Ramanujan’s Lost Notebook

  • Hannah E. BursonEmail author
Article
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Abstract

In his lost notebook, Ramanujan listed five identities related to the false theta function:
$$\begin{aligned} f(q)=\sum _{n=0}^\infty (-1)^nq^{n(n+1)/2}. \end{aligned}$$
A new combinatorial interpretation and a proof of one of these identities are given. The methods of the proof allow for new multivariate generalizations of this identity. Additionally, the same technique can be used to obtain a combinatorial interpretation of another one of the identities.

Keywords

Partitions Overpartitions False theta functions 

Mathematics Subject Classification

Primary 05A17 Secondary 05A19 

Notes

Acknowledgements

The author would like to thank Bruce Berndt for suggesting this project, and also thank Frank Garvan for suggesting Theorem 5.1 and Dennis Eichhorn for his many helpful comments.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA

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