Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Bilateral series and Ramanujan’s radial limits


Authors: J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci
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
Published electronically: October 22, 2014
MathSciNet review: 3283638
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

References
  • George E. Andrews and F. G. Garvan, Ramanujan’s “lost” notebook. VI. The mock theta conjectures, Adv. in Math. 73 (1989), no. 2, 242–255. MR 987276, DOI https://doi.org/10.1016/0001-8708%2889%2990070-4
  • Nathan J. Fine, Basic hypergeometric series and applications, Mathematical Surveys and Monographs, vol. 27, American Mathematical Society, Providence, RI, 1988. With a foreword by George E. Andrews. MR 956465
  • A. Folsom, K. Ono, and R. Rhoades, $q$-series and quantum modular forms, submitted for publication.
  • A. Folsom, K. Ono, and R. Rhoades, Ramanujan’s radial limits, submitted for publication.
  • George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
  • B. Gordon and R. McIntosh, A survey of the classical mock theta functions, Partitions, $q$-series, and modular forms, Dev. Math. 23, Springer-Verlag, New York, 2012, 95–244.
  • M. Griffin, K. Ono, and L. Rolen Ramanujan’s mock theta functions, Proceedings of the National Academy of Sciences, USA, 110, No. 15 (2013), pages 5765-5768.
  • Dean Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), no. 3, 639–660. MR 969247, DOI https://doi.org/10.1007/BF01394279
  • Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 244, Springer-Verlag, New York-Berlin, 1981. MR 648603
  • 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
  • G. N. Watson, The Mock Theta Functions (2), Proc. London Math. Soc. (2) 42 (1936), no. 4, 274–304. MR 1577032, DOI https://doi.org/10.1112/plms/s2-42.1.274
  • D. Zagier, Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann), Séminaire Bourbaki Vol. 2007/2008, Astérisque No. 326 (2009), Exp. No. 986, vii–viii, 143–164 (2010). MR2695321 (2011k:11049).
  • S. P. Zwegers, Mock $\theta $-functions and real analytic modular forms, $q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 269–277. MR 1874536, DOI https://doi.org/10.1090/conm/291/04907
  • S. Zwegers, Mock theta functions, Ph.D. Thesis (Advisor: D. Zagier), Universiteit Utrecht, 2002.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11F37, 33D15

Retrieve articles in all journals with MSC (2010): 11F37, 33D15


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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.