A generalization of a beautiful q-series identity found in the unorganized portion of Ramanujan’s second and third notebooks is obtained. As a consequence, we derive a three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews et al., which itself generalizes the famous result of Fokkink et al. This three-parameter identity also leads to several new weighted partition identities as well as a natural proof of a recent result of Garvan. This natural proof gives interesting number-theoretic information along the way. We also obtain a new result consisting of an infinite series involving a special case of Fine’s function F(a, b; t), namely \(F(0,q^n;cq^n)\). For \(c=1\), this gives Andrews’ famous identity for \(spt (n)\), whereas for \(c=-1, 0\) and q, it unravels new relations that the divisor function d(n) has with other partition-theoretic functions such as the largest parts function \(lpt (n)\).

中文翻译：

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三参数q系列恒等式的分区含义

获得了在拉马努詹第二和第三本笔记本的无组织部分中找到的美丽q系列身份的概括。结果，我们得出了一个三参数身份，这是分区理论信息的丰富来源。尤其是，我们使用这种身份来获得对Andrews等人最近结果的概括，该结果本身也可以概括Fokkink等人的著名结果。此三参数标识还导致了几个新的加权分区标识以及Garvan最近的结果的自然证明。这种自然的证明提供了有趣的数论信息。我们还获得了一个新的结果，该结果包括一个涉及Fine函数F（a，  b; t），即\（F（0，q ^ n; cq ^ n）\）。对于\（c = 1 \），这给出了安德鲁斯对于\（spt（n）\）的著名身份，而对于\（c = -1，0 \）和q，则解开了除数函数d（n）具有其他分区理论函数，例如最大部分函数\（lpt（n）\）。