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Mock and mixed mock modular forms in the lower half-plane. (English) Zbl 1417.11062

Summary: We study mock and mixed mock modular forms in the lower half-plane. In particular, our results apply to Zwegers’ three-variable mock Jacobi form \({\mu(u,v;\tau)}\), three-variable generalizations of the universal mock modular partition rank generating function, and the quantum and mock modular strongly unimodal sequence rank generating function. We do not rely upon the analytic properties of these functions; we establish our results concisely using the theory of \(q\)-hypergeometric series and partial theta functions. We extend related results of Ramanujan, Hikami, and prior work of the author with Bringmann and Rhoades, and also incorporate more recent aspects of the theory pertaining to quantum modular forms and the behavior of these functions at rational numbers when viewed as functions of \({\tau}\) (or equivalently, at roots of unity when viewed as functions of \(q\)).

MSC:

11F37 Forms of half-integer weight; nonholomorphic modular forms
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11F27 Theta series; Weil representation; theta correspondences
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[1] K. Alladi, A partial theta identity of Ramanujan and its number-theoretic interpretation, Ramanujan J. 20 (2009), 329-339. · Zbl 1225.05025 · doi:10.1007/s11139-009-9177-x
[2] K. Alladi, A combinatorial study and comparison of partial theta identities of Andrews and Ramanujan, Ramanujan J. 23 (2010), 227-241. · Zbl 1218.05017 · doi:10.1007/s11139-009-9188-7
[3] G. Andrews, An introduction to Ramanujan’s “lost” notebook, Amer. Math. Monthly 86 (1979), 89-108. · Zbl 0401.01003 · doi:10.2307/2321943
[4] G. Andrews, Partitions: Yesterday and Today, New Zealand Mathematical Society, Wellington, 1979. · Zbl 0464.10009
[5] G. Andrews, Ramanujan’s “lost” notebook. I. Partial \[\theta\] θ functions, Adv. Math. 41 (1981), 137-172. · Zbl 0477.33001 · doi:10.1016/0001-8708(81)90013-X
[6] G. Andrews and B. Berndt, Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005. · Zbl 1075.11001
[7] G. Andrews and B. Berndt, Ramanujan’s Lost Notebook, Part II, Springer, New York, 2009. · Zbl 1180.11001
[8] B. Berndt and A. Yee, Combinatorial proofs of identities in Ramanujan’s lost notebook associated with the Rogers-Fine identity and false theta functions, Ann. Comb. 7 (2003), 409-423. · Zbl 1037.05005 · doi:10.1007/s00026-003-0194-y
[9] B. Berndt, B. Kim, and A. Yee, Ramanujan’s lost notebook: combinatorial proofs of identities associated with Heine’s transformation or partial theta functions, J. Combin. Theory Ser. A 117 (2010), 857-973. · Zbl 1227.05053 · doi:10.1016/j.jcta.2009.07.005
[10] K. Bringmann, A. Folsom, and R.C. Rhoades, Partial theta functions and mock modular forms as q-hypergeometric series, Ramanujan J., special issue Ramanujan’s 125th birthday, 29 (2012), 295-310. · Zbl 1283.11077
[11] K. Bringmann and K. Ono, Dyson’s ranks and Maass forms, Ann. of Math. (2) 171 (2010), 419-449. · Zbl 1277.11096 · doi:10.4007/annals.2010.171.419
[12] K. Bringmann and L. Rolen, Radial limits of mock theta functions, Res. Math. Sci. 2 (2015), 2-17. · Zbl 1379.11048 · doi:10.1186/s40687-015-0035-8
[13] J.H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), 45-90. · Zbl 1088.11030 · doi:10.1215/S0012-7094-04-12513-8
[14] J. Bryson, S. Pitman, K. Ono, and R.C. Rhoades, Unimodal sequences and quantum mock modular forms, Proc. Natl. Acad. Sci. USA 109 (2012), 16063-16067. · doi:10.1073/pnas.1211964109
[15] Y-S Choi, The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24 (2011), 345-386. · Zbl 1225.33019 · doi:10.1007/s11139-010-9269-7
[16] A. Dabholkar, S. Murthy, and D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074. · Zbl 1088.11030
[17] N.J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys and Monographs, 27, American Mathematical Society, Providence, RI, 1988. · Zbl 0647.05004
[18] A. Folsom, K. Ono, and R.C. Rhoades, Mock theta functions and quantum modular forms, Forum Math. Pi 1 (2013), e2, 27 p. · Zbl 1294.11083
[19] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. · Zbl 0695.33001
[20] B. Gordon and R. McIntosh, A survey of classical mock theta functions, Dev. Math., 23, Springer, New York, 2012, 95-144. · Zbl 1246.33006
[21] K. Hikami, Mock (false) theta functions as quantum invariants, Regul. Chaotic Dyn. 10 (2005), 509-530. · Zbl 1133.57301 · doi:10.1070/RD2005v010n04ABEH000328
[22] M-J Jang and S. Löbrich, Radial Limits of the Universal Mock Theta Function \[g_3\] g3, Proc. Amer. Math. Soc., to appear. · Zbl 1421.11033
[23] K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Current developments in mathematics, Int. Press, Somerville, MA, 2008, 347-454. · Zbl 1229.11074
[24] L. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. (2) 16 (1917), 315-336. · JFM 46.0109.01
[25] G. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc. 2 (1936), 55-80. · Zbl 0013.11502 · doi:10.1112/jlms/s1-11.1.55
[26] D. Zagier, Quantum modular forms, In Quanta of Maths: Conference in honor of Alain Connes, Clay Math. Proc. 11 (2010), Amer. Math. Soc., Providence, RI, 659-675. · Zbl 1294.11084
[27] S.P. Zwegers, Mock \[\theta\] θ-functions and real analytic modular forms, \[q\] q-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000), Contemp. Math., 291, American Mathematical Society, Providence, RI, 2001. 269-277. · Zbl 1044.11029
[28] S. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 2002. · Zbl 1194.11058
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