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Radial limits of the universal mock theta function $ g_3$


Authors: Min-Joo Jang and Steffen Löbrich
Journal: Proc. Amer. Math. Soc. 145 (2017), 925-935
MSC (2010): Primary 11F99
DOI: https://doi.org/10.1090/proc/13065
Published electronically: November 28, 2016
MathSciNet review: 3589294
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Abstract: Referring to Ramanujan's original definition of a mock theta function, Rhoades asked for explicit formulas for radial limits of the universal mock theta functions $ g_2$ and $ g_3$. Recently, Bringmann and Rolen found such formulas for specializations of $ g_2$. Here we treat the case of $ g_3$, generalizing radial limit results for the rank generating function of Folsom, Ono, and Rhoades. Furthermore, we find expressions for radial limits of fifth order mock theta functions different from those of Bajpai, Kimport, Liang, Ma, and Ricci.


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  • [1] George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook. Part II, Springer, New York, 2009. MR 2474043 (2010f:11002)
  • [2] George E. Andrews and Dean Hickerson, Ramanujan's ``lost'' notebook. VII. The sixth order mock theta functions, Adv. Math. 89 (1991), no. 1, 60-105. MR 1123099 (92i:11027), https://doi.org/10.1016/0001-8708(91)90083-J
  • [3] J. Bajpai, S. Kimport, J. Liang, D. Ma, and J. Ricci, Bilateral series and Ramanujan's radial limits, Proc. Amer. Math. Soc. 143 (2015), no. 2, 479-492. MR 3283638, https://doi.org/10.1090/S0002-9939-2014-12249-0
  • [4] Kathrin Bringmann, Amanda Folsom, and Robert C. Rhoades, Partial theta functions and mock modular forms as $ q$-hypergeometric series, Ramanujan J. 29 (2012), no. 1-3, 295-310. MR 2994103, https://doi.org/10.1007/s11139-012-9370-1
  • [5] Kathrin Bringmann and Larry Rolen, Half-integral weight Eichler integrals and quantum modular forms, J. Number Theory 161 (2016), 240-254. MR 3435727, https://doi.org/10.1016/j.jnt.2015.03.001
  • [6] Kathrin Bringmann and Larry Rolen, Radial limits of mock theta functions, Res. Math. Sci. 2 (2015), Art. 17, 18. MR 3392042, https://doi.org/10.1186/s40687-015-0035-8
  • [7] D. Choi, S. Lim, and R. Rhoades, Mock modular forms and quantum modular forms (preprint).
  • [8] Amanda Folsom, Ken Ono, and Robert C. Rhoades, Mock theta functions and quantum modular forms, Forum Math. Pi 1 (2013), e2, 27. MR 3141412
  • [9] Basil Gordon and Richard J. McIntosh, A survey of classical mock theta functions, Partitions, $ q$-series, and modular forms, Dev. Math., vol. 23, Springer, New York, 2012, pp. 95-144. MR 3051186, https://doi.org/10.1007/978-1-4614-0028-8_9
  • [10] Michael Griffin, Ken Ono, and Larry Rolen, Ramanujan's mock theta functions, Proc. Natl. Acad. Sci. USA 110 (2013), no. 15, 5765-5768. MR 3065809, https://doi.org/10.1073/pnas.1300345110
  • [11] Soon-Yi Kang, Mock Jacobi forms in basic hypergeometric series, Compos. Math. 145 (2009), no. 3, 553-565. MR 2507741 (2010f:33022), https://doi.org/10.1112/S0010437X09004060
  • [12] E. Mortenson, Ramanujan's radial limits and mixed mock modular bilateral $ q$-hypergeometric series, ArXiv e-prints (2013), available at 1309.4162. Provided by the SAO/NASA Astrophysics Data System.
  • [13] Robert C. Rhoades, On Ramanujan's definition of mock theta function, Proc. Natl. Acad. Sci. USA 110 (2013), no. 19, 7592-7594. MR 3067051, https://doi.org/10.1073/pnas.1301046110
  • [14] 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)
  • [15] Wadim Zudilin, On three theorems of Folsom, Ono and Rhoades, Proc. Amer. Math. Soc. 143 (2015), no. 4, 1471-1476. MR 3314062, https://doi.org/10.1090/S0002-9939-2014-12364-1
  • [16] S. Zwegers, Mock Theta Functions, ArXiv e-prints (2008), available at 0807.4834. Provided by the SAO/NASA Astrophysics Data System.

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Additional Information

Min-Joo Jang
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: min-joo.jang@uni-koeln.de

Steffen Löbrich
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: steffen.loebrich@uni-koeln.de

DOI: https://doi.org/10.1090/proc/13065
Received by editor(s): April 21, 2015
Received by editor(s) in revised form: December 10, 2015
Published electronically: November 28, 2016
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society

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