Abstract
The modularity of the partition-generating function has many important consequences: for example, asymptotics and congruences for . In a pair of articles, Bringmann and Ono [11], [12] connected the rank, a partition statistic introduced by Dyson [18], to weak Maass forms, a new class of functions that are related to modular forms and that were first considered in [14]. Here, we take a further step toward understanding how weak Maass forms arise from interesting partition statistics by placing certain -marked Durfee symbols introduced by Andrews [1] into the framework of weak Maass forms. To do this, we construct a new class of functions that we call quasi-weak Maass forms because they have quasi-modular forms as components. As an application, we prove two conjectures of Andrews [1, Conjectures 11, 13]. It seems that this new class of functions will play an important role in better understanding weak Maass forms of higher weight themselves and also their derivatives. As a side product, we introduce a new method that enables us to prove transformation laws for generating functions over incomplete lattices
Citation
Kathrin Bringmann. "On the explicit construction of higher deformations of partition statistics." Duke Math. J. 144 (2) 195 - 233, 15 August 2008. https://doi.org/10.1215/00127094-2008-035
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