Quantum modularity of mock theta functions of order 2

Soon-Yi Kang


In [9], we computed shadows of the second order mock theta functions and showed that they are essentially same with the shadow of a mock theta function related to the Mathieu moonshine phenomenon. In this paper, we further survey the second order mock theta functions on their quantum modularity and their behavior in the lower half plane.


basic hypergeometric series, second order mock theta functions, quantum modular forms

Full Text:



G. E. Andrews, Mordell integrals and Ramanujan’s “lost”notebook, Analytic number theory, (Philadelphia, Pa., 1980), pp. 1018, Lecture Notes in Math., 899, Springer, Berlin-New York, 1981.

K. Bringmann, A. Folsom, and R. C. Rhoades, Partial theta functions and mock modular forms as q-hypergeometric series, Ramanujan J. 29 (2012), 295-310.

K. Bringmann and L. Rolen, Radial limits of mock theta functions, Res. Math. Sci. 2 (2015), 2–17.

A. Folsom, S. Garthwaite, S.-Y. Kang, H. Swisher and S. Treneer, Quantum mock modular forms arising from eta-theta fnctions, Res. Number Theory 2 (2016).

A. Folsom, K. Ono and R. C. Rhoades, Mock theta functions and quantum modular forms, Forum Math. Pi 1 (2013), e2, 27p.

B. Gordon and R. J. McIntosh, A survey of classical mock theta functions, Par- titions, q-Series, and Modular Forms, pp. 95–144, Springer, Berlin, 2012.

K. Hikami, Mock (false) theta functions as quantum invariants, Regul. Chaotic Dyn. 10 (2005), 509–530.

S.-Y. Kang, Mock Jacobi forms in basic hypergeometric series, Compos. Math. 145 (3) (2009), 553–565.

S.-Y. Kang and H. Swisher, Mock theta functions of order 2 and their shadow computations, Bull. Korean Math. Soc. (to appear).

R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3- manifolds, Asian J. Math. 3 (1999) 93–108.

R. J. McIntosh, Second order mock theta functions, Canad. Math. Bull. 50 (2) (2007), 284–290.

R. J. McIntosh, The H and K family of mock theta functions, Canad. J. Math. 64 (2012), 935–960.

R. J. McIntosh, On the universal mock theta function g2 and Zwegers’ μ- function, Proceedings of Alladi60 Conference (2016), http://qseries.org/ alladi60/talks/mcintosh/

K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Current developments in mathematics 2008 (2009), 347–454.

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.

D. Zagier, Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), S`eminaire Bourbaki, Vol. 2007/2008. Astrisque No. 326 (2009), Exp. No. 986, viiviii, 143164 (2010).

D. Zagier, Quantum Modular Forms, In Quanta of Maths: Conference in honor of Alain Connes, Clay Math. Proc. 11 (2010), Amer. Math. Soc., Providence, RI, 659–675.

S. P. Zwegers, Mock Theta Functions Thesis, Utrecht, 2002, https://dspace. library.uu.nl/bitstream/handle/1874/878/full.pdf?sequence=11.

S. P. Zwegers, Multivariable Appell functions, 2010 http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid= 5753340C9807A361CBD13F25FD0DEA7D?doi= rep1&type=pdf.

DOI: http://dx.doi.org/10.11568/kjm.2017.25.1.87


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr