Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)

  • [1] George E. Andrews, Bailey chains and generalized Lambert series. I. Four identities of Ramanujan, Illinois J. Math. 36 (1992), no. 2, 251-274. MR 1156626 (93i:11020)
  • [2] George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook. Part II, Springer, New York, 2009. MR 2474043 (2010f:11002)
  • [3] George E. Andrews, Freeman J. Dyson, and Dean Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), no. 3, 391-407. MR 928489 (89f:11071), https://doi.org/10.1007/BF01388778
  • [4] George E. Andrews, Robert C. Rhoades, and Sander P. Zwegers, Modularity of the concave composition generating function, Algebra Number Theory 7 (2013), no. 9, 2103-2139. MR 3152010, https://doi.org/10.2140/ant.2013.7.2103
  • [5] Kathrin Bringmann, Amanda Folsom, and Robert C. Rhoades, Unimodal sequences and ``strange'' functions: a family of quantum modular forms, Pacific J. Math. 274 (2015), no. 1, 1-25. MR 3320867, https://doi.org/10.2140/pjm.2015.274.1
  • [6] K. Bringmann and L. Rolen, Half-integral weight Eichler integrals and quantum modular forms, J. Number Theory, to appear.
  • [7] R. Bruggeman, Quantum Maass forms, The Conference on $ L$-Functions, World Sci. Publ., Hackensack, NJ, 2007, pp. 1-15. MR 2310286 (2008g:11076)
  • [8] Jennifer Bryson, Ken Ono, Sarah Pitman, and Robert C. Rhoades, Unimodal sequences and quantum and mock modular forms, Proc. Natl. Acad. Sci. USA 109 (2012), no. 40, 16063-16067. MR 2994899, https://doi.org/10.1073/pnas.1211964109
  • [9] D. Choi, S. Lim, and R. C. Rhoades, Mock modular forms and quantum modular forms, Proc. Amer. Math. Soc., to appear.
  • [10] Amanda Folsom, Ken Ono, and Robert C. Rhoades, Mock theta functions and quantum modular forms, Forum Math. Pi 1 (2013), e2, 27. MR 3141412
  • [11] Amanda Folsom, Ken Ono, and Robert C. Rhoades, Ramanujan's radial limits, Ramanujan 125, Contemp. Math., vol. 627, Amer. Math. Soc., Providence, RI, 2014, pp. 91-102. MR 3307493, https://doi.org/10.1090/conm/627/12534
  • [12] Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and $ t$-cores, Invent. Math. 101 (1990), no. 1, 1-17. MR 1055707 (91h:11106), https://doi.org/10.1007/BF01231493
  • [13] Kazuhiro Hikami, Quantum invariants, modular forms, and lattice points. II, J. Math. Phys. 47 (2006), no. 10, 102301, 32. MR 2268852 (2008f:58023), https://doi.org/10.1063/1.2349484
  • [14] Kazuhiro Hikami and Jeremy Lovejoy, Torus knots and quantum modular forms, Res. Math. Sci. 2 (2015), Art. 2, 15. MR 3324400, https://doi.org/10.1186/s40687-014-0016-3
  • [15] Dean R. Hickerson and Eric T. Mortenson, Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I, Proc. Lond. Math. Soc. (3) 109 (2014), no. 2, 382-422. MR 3254929, https://doi.org/10.1112/plms/pdu007
  • [16] Marvin Knopp and Henok Mawi, Eichler cohomology theorem for automorphic forms of small weights, Proc. Amer. Math. Soc. 138 (2010), no. 2, 395-404. MR 2557156 (2010i:11055), https://doi.org/10.1090/S0002-9939-09-10070-9
  • [17] Jeremy Lovejoy, A Bailey lattice, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1507-1516 (electronic). MR 2053359 (2005b:33017), https://doi.org/10.1090/S0002-9939-03-07247-2
  • [18] Jeremy Lovejoy, Overpartitions and real quadratic fields, J. Number Theory 106 (2004), no. 1, 178-186. MR 2049600 (2005b:11164), https://doi.org/10.1016/j.jnt.2003.12.014
  • [19] Jeremy Lovejoy, Ramanujan-type partial theta identities and conjugate Bailey pairs, Ramanujan J. 29 (2012), no. 1-3, 51-67. MR 2994089, https://doi.org/10.1007/s11139-011-9356-4
  • [20] Richard J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2007), no. 2, 284-290. MR 2317449 (2008d:11017), https://doi.org/10.4153/CMB-2007-028-9
  • [21] Eric T. Mortenson, On the dual nature of partial theta functions and Appell-Lerch sums, Adv. Math. 264 (2014), 236-260. MR 3250284, https://doi.org/10.1016/j.aim.2014.07.018
  • [22] E. T. Mortenson, Ramanujan's radial limits and mixed mock modular bilateral $ q$-hypergeometric series, Proc. Edinburgh Math. Soc., to appear.
  • [23] Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $ q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489 (2005c:11053)
  • [24] Robert C. Rhoades, Asymptotics for the number of strongly unimodal sequences, Int. Math. Res. Not. IMRN 3 (2014), 700-719. MR 3163564
  • [25] Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440-481. MR 0332663 (48 #10989)
  • [26] L. J. Slater, Further identies of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147-167. MR 0049225 (14,138e)
  • [27] Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40 (2001), no. 5, 945-960. MR 1860536 (2002g:11055), https://doi.org/10.1016/S0040-9383(00)00005-7
  • [28] Don Zagier, Quantum modular forms, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 659-675. MR 2757599 (2012a:11066)
  • [29] S. Zwegers, Mock Theta Functions, Ph.D. Thesis, Universiteit Utrecht (2002).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F33, 11F37, 33D15

Retrieve articles in all journals with MSC (2010): 11F33, 11F37, 33D15


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

American Mathematical Society