Duke Mathematical Journal
- Duke Math. J.
- Volume 144, Number 2 (2008), 195-233.
On the explicit construction of higher deformations of partition statistics
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 ,  connected the rank, a partition statistic introduced by Dyson , to weak Maass forms, a new class of functions that are related to modular forms and that were first considered in . 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  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
Duke Math. J. Volume 144, Number 2 (2008), 195-233.
First available in Project Euclid: 14 August 2008
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11P82: Analytic theory of partitions
Secondary: 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]
Bringmann, Kathrin. On the explicit construction of higher deformations of partition statistics. Duke Math. J. 144 (2008), no. 2, 195--233. doi:10.1215/00127094-2008-035. https://projecteuclid.org/euclid.dmj/1218716298