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    Folsom, Amanda Garthwaite, Sharon Kang, Soon-Yi Swisher, Holly and Treneer, Stephanie 2016. Quantum mock modular forms arising from eta–theta functions. Research in Number Theory, Vol. 2, Issue. 1,


    Hickerson, Dean and Mortenson, Eric 2016. Dyson’s ranks and Appell–Lerch sums. Mathematische Annalen,


    Bringmann, Kathrin and Rolen, Larry 2015. Radial limits of mock theta functions. Research in the Mathematical Sciences, Vol. 2, Issue. 1,


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    Lovejoy, Jeremy and Osburn, Robert 2013. The Bailey chain and mock theta functions. Advances in Mathematics, Vol. 238, p. 442.


    Bringmann, Kathrin Folsom, Amanda and Rhoades, Robert C. 2012. Partial theta functions and mock modular forms as q-hypergeometric series. The Ramanujan Journal, Vol. 29, Issue. 1-3, p. 295.


    Kim, Byungchan 2012. Periodicity of signs of Fourier coefficients of eta-quotients. Journal of Mathematical Analysis and Applications, Vol. 385, Issue. 2, p. 998.


    Choi, Youn-Seo 2011. The basic bilateral hypergeometric series and the mock theta functions. The Ramanujan Journal, Vol. 24, Issue. 3, p. 345.


    Choie, YoungJu and Lim, Subong 2010. The heat operator and mock Jacobi forms. The Ramanujan Journal, Vol. 22, Issue. 2, p. 209.


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Mock Jacobi forms in basic hypergeometric series

  • Soon-Yi Kang (a1)
  • DOI: http://dx.doi.org/10.1112/S0010437X09004060
  • Published online: 01 May 2009
Abstract
Abstract

We show that some q-series such as universal mock theta functions are linear sums of theta quotients and mock Jacobi forms of weight 1/2, which become holomorphic parts of real analytic modular forms when they are restricted to torsion points and multiplied by suitable powers of q. We also prove that certain linear sums of q-series are weakly holomorphic modular forms of weight 1/2 due to annihilation of mock Jacobi forms or completion by mock Jacobi forms. As an application, we obtain a relation between the rank and crank of a partition.

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[2]G. E. Andrews and F. G. Garvan , Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N. S.) 18 (1988), 167171.

[3]A. O. L. Atkin and F. G. Garvan , Relations between the ranks and cranks of partitions (Rankin memorial issues), Ramanujan J. 7 (2003), 343366.

[5]K. Bringmann and K. Ono , The f(q) mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), 243266.

[7]K. Bringmann , K. Ono and R. C. Rhoades , Eulerian series as modular forms, J. Amer. Math. Soc. 21 (2008), 10851104.

[10]M. Eichler and D. Zagier , The theory of Jacobi forms, Progress in Mathematics, vol. 55 (Birkhauser, Basel, 1985).

[13]D. Hickerson , A proof of the mock theta conjectures, Invent. Math. 94 (1988), 639660.

[14]D. Hickerson , On the seventh order mock theta functions, Invent. Math. 94 (1988), 661677.

[15]S.-Y. Kang , Generalizations of Ramanujan’s reciprocity theorem and their applications, J. London Math. Soc. (2) 75 (2007), 1834.

[18]S. P. Zwegers , Mock ϑ-functions and real analytic modular forms, in q-series with applications to combinatorics, number theory, and physics, University of Illinois at Urbana-Champaign, October 26–28, 2000, Contemporary Mathematics, vol. 291 (American Mathematical Society, Providence, RI, 2001), 269277.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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