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Odd-balanced unimodal sequences and related functions: parity, mock modularity and quantum modularity


Authors: Byungchan Kim, Subong Lim and Jeremy Lovejoy
Journal: Proc. Amer. Math. Soc. 144 (2016), 3687-3700
MSC (2010): Primary 11F33, 11F37, 33D15
DOI: https://doi.org/10.1090/proc/13027
Published electronically: March 17, 2016
MathSciNet review: 3513531
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Abstract: We define odd-balanced unimodal sequences and show that their generating function $ \mathcal {V}(x,q)$ has the same remarkable features as the generating function for strongly unimodal sequences $ U(x,q)$. In particular, we discuss (mixed) mock modularity, quantum modularity, and congruences modulo $ 2$ and $ 4$. We also study two related functions which share some of the properties of $ U(x,q)$ and $ \mathcal {V}(x,q)$.


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Additional Information

Byungchan Kim
Affiliation: School of Liberal Arts, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul 01811, Korea
Email: bkim4@seoultech.ac.kr

Subong Lim
Affiliation: Department of Mathematics Education, Sungkyunkwan University, 25-2, Sungkyunkwan-ro, Jongno-gu, Seoul 03063, Republic of Korea
Email: subong@skku.edu

Jeremy Lovejoy
Affiliation: CNRS, LIAFA, Université Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, France
Email: lovejoy@math.cnrs.fr

DOI: https://doi.org/10.1090/proc/13027
Keywords: Unimodal sequences, rank, quantum modular forms, mock theta functions, congruences
Received by editor(s): March 13, 2015
Received by editor(s) in revised form: November 3, 2015
Published electronically: March 17, 2016
Additional Notes: This research was supported by the International Research & Development Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology(MEST) of Korea (NRF-2014K1A3A1A21000358), and the STAR program number 32142ZM. The second author was supported by Samsung Science and Technology Foundation under Project SSTF-BA1301-11.
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2016 American Mathematical Society