论文标题
Ramanujan的五个Q系列身份和未开发的加权分区身份的概括
Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities
论文作者
论文摘要
Ramanujan在一部分中记录了五个有趣的Q系列身份，这些身份与他的第二张笔记本的其他章节不像其他章节那样安排。这五个身份似乎没有引起足够的关注。最近，Dixit和第三作者发现了上述五个身份之一。从他们的广义身份中，他们能够得出这些Q系列身份的最后三个，但没有确定前两个。在本文中，我们得出了对Dixit的主要身份和第三作者的一般概括，我们成功地推断出Ramanujan的所有五个Q系列身份。除此之外，我们还从我们的广义身份中建立了一些有趣的加权分区身份。在1980年代中期，Bressoud和Subbarao发现了一个有趣的身份，将广义除数函数与加权分区函数联系起来，它们通过纯粹的组合论证证明了这一点。令人惊讶的是，我们从上述五个Ramanujan的五个Q系列身份的第四个身份开始，找到了Bressoud和Subbarao身份的概括的分析证明。
Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From their generalized identity, they were able to derive the last three of these q-series identities, but didn't establish the first two. In the present article, we derive a one-variable generalization of the main identity of Dixit and the third author from which we successfully deduce all the five q-series identities of Ramanujan. In addition to this, we also establish a few interesting weighted partition identities from our generalized identity. In the mid 1980's, Bressoud and Subbarao found an interesting identity connecting the generalized divisor function with a weighted partition function, which they proved by means of a purely combinatorial argument. Quite surprisingly, we found an analytic proof for a generalization of the identity of Bressoud and Subbarao, starting from the fourth identity of the aforementioned five q-series identities of Ramanujan.
