Eulerian series as Modular forms revisited
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- by Eric T. Mortenson PDF
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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.References
- George E. Andrews, Mordell integrals and Ramanujan’s “lost” notebook, Analytic number theory (Philadelphia, Pa., 1980) Lecture Notes in Math., vol. 899, Springer, Berlin-New York, 1981, pp. 10–18. MR 654518
- George E. Andrews and Bruce C. Berndt, Ramanujan’s lost notebook. Part I, Springer, New York, 2005. MR 2135178
- 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 10.1016/0001-8708(89)90070-4
- 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, DOI 10.1016/0001-8708(91)90083-J
- 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, DOI 10.1090/S0894-0347-07-00587-5
- 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, DOI 10.1017/CBO9780511526251
- 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, DOI 10.1007/978-1-4614-0028-8_{9}
- Dean Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), no. 3, 639–660. MR 969247, DOI 10.1007/BF01394279
- Dean Hickerson, On the seventh order mock theta functions, Invent. Math. 94 (1988), no. 3, 661–677. MR 969248, DOI 10.1007/BF01394280
- 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, DOI 10.1112/plms/pdu007
- Soon-Yi Kang, Mock Jacobi forms in basic hypergeometric series, Compos. Math. 145 (2009), no. 3, 553–565. MR 2507741, DOI 10.1112/S0010437X09004060
- Jeremy Lovejoy and Robert Osburn, The Bailey chain and mock theta functions, Adv. Math. 238 (2013), 442–458. MR 3033639, DOI 10.1016/j.aim.2013.02.005
- Jeremy Lovejoy and Robert Osburn, $q$-hypergeometric double sums as mock theta functions, Pacific J. Math. 264 (2013), no. 1, 151–162. MR 3079764, DOI 10.2140/pjm.2013.264.151
- J. Lovejoy, R. Osburn, On two 10th order mock theta identities, Ramanujan Journal, to appear, arxiv:1209.2315.
- Eric T. Mortenson, On the dual nature of partial theta functions and Appell-Lerch sums, Adv. Math. 264 (2014), 236–260. MR 3250284, DOI 10.1016/j.aim.2014.07.018
- Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735
- S. P. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 2002.
Additional Information
- Eric T. Mortenson
- Affiliation: Max-Planck-Institut für Mathematik, Vitvatsgasse 7, 53111 Bonn, Germany
- Email: etmortenson@gmail.com
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2379-2385
- MSC (2010): Primary 11B65, 11F11, 11F27
- DOI: https://doi.org/10.1090/S0002-9939-2015-12451-3
- MathSciNet review: 3326020