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

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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.

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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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Correspondence to Hannah E. Burson.

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Dedicated to Professor George Andrews on the occasion of his eightieth birthday.

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Burson, H.E. A Bijective Proof of a False Theta Function Identity from Ramanujan’s Lost Notebook. Ann. Comb. 23, 579–588 (2019). https://doi.org/10.1007/s00026-019-00454-7

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  • DOI: https://doi.org/10.1007/s00026-019-00454-7

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