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Eulerian series as Modular forms revisited

Author: Eric T. Mortenson
Journal: Proc. Amer. Math. Soc. 143 (2015), 2379-2385
MSC (2010): Primary 11B65, 11F11, 11F27
Published electronically: February 4, 2015
MathSciNet review: 3326020
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Abstract: Recently, Bringmann, Ono, and Rhoades employed harmonic weak Maass forms to prove results on Eulerian series as modular forms. By changing the setting to Appell-Lerch sums, we shorten the proof of one of their main theorems. In addition we discuss connections to recent work of Kang.

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  • [1] George E. Andrews, Mordell integrals and Ramanujan's ``lost'' notebook, Analytic number theory (Philadelphia, Pa., 1980) Lecture Notes in Math., vol. 899, Springer, Berlin, 1981, pp. 10-18. MR 654518 (83m:33004)
  • [2] George E. Andrews and Bruce C. Berndt, Ramanujan's lost notebook. Part I, Springer, New York, 2005. MR 2135178 (2005m:11001)
  • [3] 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 (90d:11115),
  • [4] 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),
  • [5] Kathrin Bringmann, Ken Ono, and Robert C. Rhoades, Eulerian series as modular forms, J. Amer. Math. Soc. 21 (2008), no. 4, 1085-1104. MR 2425181 (2010a:11078),
  • [6] George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719 (2006d:33028)
  • [7] 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,
  • [8] Dean Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), no. 3, 639-660. MR 969247 (90f:11028a),
  • [9] Dean Hickerson, On the seventh order mock theta functions, Invent. Math. 94 (1988), no. 3, 661-677. MR 969248 (90f:11028b),
  • [10] Dean R. Hickerson and Eric T. Mortenson, Hecke-type double sums, Appell-Lerch sums, and mock theta functions, I, Proc. Lond. Math. Soc. (3) 109 (2014), no. 2, 382-422. MR 3254929,
  • [11] Soon-Yi Kang, Mock Jacobi forms in basic hypergeometric series, Compos. Math. 145 (2009), no. 3, 553-565. MR 2507741 (2010f:33022),
  • [12] Jeremy Lovejoy and Robert Osburn, The Bailey chain and mock theta functions, Adv. Math. 238 (2013), 442-458. MR 3033639,
  • [13] Jeremy Lovejoy and Robert Osburn, $ q$-hypergeometric double sums as mock theta functions, Pacific J. Math. 264 (2013), no. 1, 151-162. MR 3079764,
  • [14] J. Lovejoy, R. Osburn, On two 10th order mock theta identities, Ramanujan Journal, to appear, arxiv:1209.2315.
  • [15] Eric T. Mortenson, On the dual nature of partial theta functions and Appell-Lerch sums, Adv. Math. 264 (2014), 236-260. MR 3250284,
  • [16] Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin, 1988. With an introduction by George E. Andrews. MR 947735 (89j:01078)
  • [17] S. P. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 2002.

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Additional Information

Eric T. Mortenson
Affiliation: Max-Planck-Institut für Mathematik, Vitvatsgasse 7, 53111 Bonn, Germany

Keywords: Appell--Lerch sums, Eulerian forms, $q$-hypergeometric series
Received by editor(s): September 23, 2013
Received by editor(s) in revised form: January 21, 2014, and January 23, 2014
Published electronically: February 4, 2015
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2015 American Mathematical Society

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