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Proceedings of the American Mathematical Society

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Bilateral series and Ramanujan’s radial limits
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by J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci PDF
Proc. Amer. Math. Soc. 143 (2015), 479-492 Request permission

Abstract:

Ramanujan’s last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called mock theta functions. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $\zeta$, \[ \lim _{q\to \zeta } \big (f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots )\big ) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by different methods, we prove similar results for all of Ramanujan’s 5th order mock theta functions. Namely, we show that each 5th order mock theta function may be related to a modular bilateral series and exploit this connection to obtain our results. We further explore other mock theta functions to which this method can be applied.
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Additional Information
  • J. Bajpai
  • Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1. Canada
  • Email: jitendra@math.ualberta.ca
  • S. Kimport
  • Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520
  • Email: susie.kimport@yale.edu
  • J. Liang
  • Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32601
  • Email: jieliang@ufl.edu
  • D. Ma
  • Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
  • Email: martin@math.arizona.edu
  • J. Ricci
  • Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
  • Email: jricci@wesleyan.edu
  • Received by editor(s): April 24, 2013
  • Published electronically: October 22, 2014
  • Additional Notes: This project is the result of participation in the 2013 Arizona Winter School.
  • Communicated by: Ken Ono
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 479-492
  • MSC (2010): Primary 11F37, 33D15
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12249-0
  • MathSciNet review: 3283638