AI 中文总结
本文针对奥利弗基于模形式理论分类出的所有作为θ函数的η商，给出其对应恒等式的初等证明，相关成果依托雅可比三重乘积恒等式展开，在数论领域具有一定意义。
AI 中文摘要
雅可比三重乘积恒等式为组合数学和数论中自然出现的许多无穷乘积生成函数提供了闭式表达式，其一个重要应用是将戴德金η函数{\u03b7}({\u03c4})表示为θ函数。利用模形式理论，奥利弗[5]对所有作为θ函数的η商进行了分类，本文给出了奥利弗所建立恒等式的初等证明。
英文摘要
The Jacobi triple product identity provides closed-form expressions for many infinite-product generating functions that arise naturally in combinatorics and number theory. A particularly important application is to Dedekind's eta function η(τ), which it expresses as a theta function. Using the theory of modular forms, Oliver [5] classified all eta quotients that are theta functions. In this paper, we present elementary proofs of the identities established by Oliver.
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