Partition Statistics and Quasiharmonic Maass Forms
Partition Statistics and Quasiharmonic Maass Forms
复制标题
DOI：
10.1093/imrn/rnn124
复制
发表时间：
2009
影响因子：
1
通讯作者：
K. Bringmann;F. Garvan;K. Mahlburg
中科院分区：
文献类型：
--
作者：
K. Bringmann;F. Garvan;K. Mahlburg
Andrews最近引入了k标记的Durfee符号，它是与Dyson秩统计矩相关的分区的推广。他用这些联系，以找到身份有关其生成功能，以及证明拉马努金型同余这些对象，并找到它们之间的关系。在本文中，我们表明，这些对象的超几何生成函数是自然的例子，准模拟θ函数，这是定义为调和马斯形式及其衍生物的全纯部分。特别地，这些生成函数可以被视为具有任意高权重的Ramanujan的模拟theta函数的类似物。我们利用自守性质证明了Durfee符号的无穷多个同余的存在性。此外，我们表明，随着k的变化，模块化的k-标记Durfee符号的情况下，k = 2正是决定。最后，我们利用这个关系，以证明存在的一般同余秩矩的一级模形式的有界重量。
Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating to their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between them. In this paper, we show that the hypergeometric generating functions for these objects are natural examples of quasimock theta functions, which are defined as the holomorphic parts of harmonic Maass forms and their derivatives. In particular, these generating functions may be viewed as analogs of Ramanujan's mock theta functions with arbitrarily high weight. We use the automorphic properties to prove the existence of infinitely many congruences for the Durfee symbols. Furthermore, we show that as k varies, the modularity of the k-marked Durfee symbols is precisely dictated by the case k = 2. Finally, we use this relation in order to prove the existence of general congruences for rank moments in terms of level one modular forms of bounded weight.