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

Kim, Byungchan; Lim, Subong; Lovejoy, Jeremy

doi:10.1090/proc/13027

Odd-balanced unimodal sequences and related functions: parity, mock modularity and quantum modularity
HTML articles powered by AMS MathViewer

by Byungchan Kim, Subong Lim and Jeremy Lovejoy

Proc. Amer. Math. Soc. 144 (2016), 3687-3700

DOI: https://doi.org/10.1090/proc/13027

Published electronically: March 17, 2016

PDF | Request permission

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

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
George E. Andrews and Bruce C. Berndt, Ramanujan’s lost notebook. Part II, Springer, New York, 2009. MR 2474043
George E. Andrews, Freeman J. Dyson, and Dean Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 (1988), no. 3, 391–407. MR 928489, DOI 10.1007/BF01388778
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, DOI 10.2140/ant.2013.7.2103
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, DOI 10.2140/pjm.2015.274.1
K. Bringmann and L. Rolen, Half-integral weight Eichler integrals and quantum modular forms, J. Number Theory, to appear.
R. Bruggeman, Quantum Maass forms, The Conference on $L$-Functions, World Sci. Publ., Hackensack, NJ, 2007, pp. 1–15. MR 2310286
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, DOI 10.1073/pnas.1211964109
D. Choi, S. Lim, and R. C. Rhoades, Mock modular forms and quantum modular forms, Proc. Amer. Math. Soc., to appear.
Amanda Folsom, Ken Ono, and Robert C. Rhoades, Mock theta functions and quantum modular forms, Forum Math. Pi 1 (2013), e2, 27. MR 3141412, DOI 10.1017/fmp.2013.3
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, DOI 10.1090/conm/627/12534
Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and $t$-cores, Invent. Math. 101 (1990), no. 1, 1–17. MR 1055707, DOI 10.1007/BF01231493
Kazuhiro Hikami, Quantum invariants, modular forms, and lattice points. II, J. Math. Phys. 47 (2006), no. 10, 102301, 32. MR 2268852, DOI 10.1063/1.2349484
Kazuhiro Hikami and Jeremy Lovejoy, Torus knots and quantum modular forms, Res. Math. Sci. 2 (2015), Art. 2, 15. MR 3324400, DOI 10.1186/s40687-014-0016-3
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, DOI 10.1112/plms/pdu007
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, DOI 10.1090/S0002-9939-09-10070-9
Jeremy Lovejoy, A Bailey lattice, Proc. Amer. Math. Soc. 132 (2004), no. 5, 1507–1516. MR 2053359, DOI 10.1090/S0002-9939-03-07247-2
Jeremy Lovejoy, Overpartitions and real quadratic fields, J. Number Theory 106 (2004), no. 1, 178–186. MR 2049600, DOI 10.1016/j.jnt.2003.12.014
Jeremy Lovejoy, Ramanujan-type partial theta identities and conjugate Bailey pairs, Ramanujan J. 29 (2012), no. 1-3, 51–67. MR 2994089, DOI 10.1007/s11139-011-9356-4
Richard J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2007), no. 2, 284–290. MR 2317449, DOI 10.4153/CMB-2007-028-9
Eric T. Mortenson, On the dual nature of partial theta functions and Appell-Lerch sums, Adv. Math. 264 (2014), 236–260. MR 3250284, DOI 10.1016/j.aim.2014.07.018
E. T. Mortenson, Ramanujan’s radial limits and mixed mock modular bilateral $q$-hypergeometric series, Proc. Edinburgh Math. Soc., to appear.
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
Robert C. Rhoades, Asymptotics for the number of strongly unimodal sequences, Int. Math. Res. Not. IMRN 3 (2014), 700–719. MR 3163564, DOI 10.1093/imrn/rns226
Goro Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97 (1973), 440–481. MR 332663, DOI 10.2307/1970831
L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167. MR 49225, DOI 10.1112/plms/s2-54.2.147
Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology 40 (2001), no. 5, 945–960. MR 1860536, DOI 10.1016/S0040-9383(00)00005-7
Don Zagier, Quantum modular forms, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 659–675. MR 2757599
S. Zwegers, Mock Theta Functions, Ph.D. Thesis, Universiteit Utrecht (2002).