A celebrated result of Bringmann and Ono shows that the combinatorial rank generating function exhibits automorphic properties after being completed by the addition of a non-holomorphic integral. Since then, automorphic properties of various related combinatorial families have been studied. Here, extending work of Andrews and Bringmann, we study general infinite families of combinatorial q-series pertaining to k-marked Durfee symbols, in which we allow additional singularities. We show that these singular combinatorial families are essentially mixed mock and quasimock modular forms, and provide their explicit non-holomorphic completions. As a special case of our work, we consider k=3, and provide an asymptotic expansion for the associated partition rank statistic, solving a special case of an open problem of Andrews.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-3-4,
author = {Amanda Folsom and Susie Kimport},
title = {Mock modular forms and singular combinatorial series},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {257-297},
zbl = {1317.11046},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-3-4}
}
Amanda Folsom; Susie Kimport. Mock modular forms and singular combinatorial series. Acta Arithmetica, Tome 161 (2013) pp. 257-297. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa159-3-4/