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.

中文翻译：

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Ramanujan遗失笔记本中虚假Theta函数身份的双证明

在他遗失的笔记本中，拉马努詹列出了与虚假theta函数相关的五个身份：$$ \ begin {aligned} f（q）= \ sum _ {n = 0} ^ \ infty（-1）^ nq ^ {n（n +1）/ 2}。\ end {aligned} $$给出了新的组合解释和这些身份之一的证明。证明方法允许对该身份进行新的多元概括。另外，可以使用相同的技术来获得对另一个身份的组合解释。