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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

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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


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
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  • S. Kimport
  • Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520
  • Email:
  • J. Liang
  • Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32601
  • Email:
  • D. Ma
  • Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
  • Email:
  • J. Ricci
  • Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
  • Email:
  • 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:
  • MathSciNet review: 3283638