Abstract
Our results investigate mock theta functions and quantum modular forms via quantum q-series identities. After Lovejoy, quantum q-series identities are such that they do not hold as an equality between power series inside the unit disc in the classical sense, but do hold at dense sets of roots of unity on the boundary. We establish several general (multivariable) quantum q-series identities and apply them to various settings involving (universal) mock theta functions. As a consequence, we surprisingly show that limiting, finite, universal mock theta functions at roots of unity for which their infinite counterparts do not converge are quantum modular. Moreover, we show that these finite limiting universal mock theta functions play key roles in (generalized) Ramanujan radial limits. A further corollary of our work reveals that the finite Kontsevich–Zagier series is a kind of “universal quantum mock theta function,” in that it may be used to evaluate odd-order Ramanujan mock theta functions at roots of unity. (We also offer a similar result for even-order mock theta functions.) Finally, to complement the notion of a quantum q-series identity and the results of this paper, we also define what we call an “antiquantum q-series identity’ and offer motivating general results with applications to third-order mock theta functions.
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References
Andrews, G.E.: Mock theta functions. In Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987), volume 49, Part 2 of Proc. Sympos. Pure Math., pp. 283–298. Amer. Math. Soc., Providence, RI (1989)
Andrews, G.E., Dyson, F.J., Hickerson, D.: Partitions and indefinite quadratic forms. Invent. Math. 91(3), 391–407 (1988)
Berndt, B.C., Dixit, A., Gupta, R.: Generalizations of the Andrews–Yee identities associated with the mock theta functions \(\omega (q)\) and \(\nu (q)\). J. Algebr. Combin. 55(4), 1031–1062 (2022)
Berndt, B.C., Rankin, R.A.: Ramanujan, volume 9 of History of Mathematics. American Mathematical Society, Providence, RI; London Mathematical Society, London. Letters and commentary (1995)
Sandro Bettin and John Brian Conrey: A reciprocity formula for a cotangent sum. Int. Math. Res. Not. IMRN 24, 5709–5726 (2013)
Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass forms and mock modular forms: theory and applications. American Mathematical Society Colloquium Publications, vol. 64. American Mathematical Society, Providence, RI (2017)
Bringmann, K., Folsom, A., Rhoades, R.C.: Partial theta functions and mock modular forms as \(q\)-hypergeometric series. Ramanujan J. 29(1–3), 295–310 (2012)
Bringmann, K., Rolen, L.: Radial limits of mock theta functions. Res. Math. Sci. 2, 18 (2015)
Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004)
Carroll, G., Corbett, J., Folsom, A., Thieu, E.: Universal mock theta functions as quantum Jacobi forms. Res. Math. Sci. 6(1), 15 (2019)
Choi, D., Lim, S., Rhoades, R.C.: Mock modular forms and quantum modular forms. Proc. Amer. Math. Soc. 144(6), 2337–2349 (2016)
Cohen, H.: \(q\)-identities for Mass waveforms. Invent. Math. 91(3), 409–422 (1988)
Duke, W.: Almost a century of answering the question: what is a mock theta function? Notices Amer. Math. Soc. 61(11), 1314–1320 (2014)
Fine, N.J.: Basic hypergeometric series and applications, volume 27 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI. With a foreword by George E. Andrews (1988)
Folsom, A.: What is \(\dots \) a mock modular form? Notices Amer. Math. Soc. 57(11), 1441–1443 (2010)
Folsom, A.: Quantum Jacobi forms in number theory, topology, and mathematical physics. Res. Math. Sci. 6(3), 34 (2019)
Folsom, A.: Asymptotics and Ramanujan’s mock theta functions: then and now. Philos. Trans. Roy. Soc. A. 378(2163) (2020)
Folsom, A.: Twisted Eisenstein series, cotangent-zeta sums, and quantum modular forms. Trans. London Math. Soc. 7(1), 33–48 (2020)
Folsom, A., Jang, M.-J., Kimport, S., Swisher, H.: Quantum modular forms and singular combinatorial series with distinct roots of unity. In Research Directions in Number Theory—Women in Numbers IV, volume 19 of Assoc. Women Math. Ser., pp. 173–195. Springer, Cham (2019)
Folsom, A., Jang, M.-J., Kimport, S., Swisher, H.: Quantum modular forms and singular combinatorial series with repeated roots of unity. Acta Arith 194(4), 393–421 (2020)
Folsom, A., Ki, C., Vu, Y.N.T., Bowen, Y.: combinatorial quantum modular forms. J. Number Theory 170, 315–346 (2017)
Folsom, A., Ono, K., Rhoades, R.C.: Mock theta functions and quantum modular forms. Forum Math. Pi, 1:e2, 27 (2013)
Folsom, A., Pratt, E., Solomon, N., Tawfeek, A.R.: Quantum Jacobi forms and sums of tails identities. Res. Number Theory 8(1), 24 (2022)
Gasper, G., Rahman, M.: Basic Hypergeometric Series, volume 96 of Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge. With a foreword by Richard Askey (2004)
Gordon, B., McIntosh, R.J.: A survey of classical mock theta functions. In: Partitions, \(q\)-Series, and Modular Forms, volume 23 of Dev. Math., pp. 95–144. Springer, New York (2012)
Hikami, K., Lovejoy, J.: Torus knots and quantum modular forms. Res. Math. Sci. 2, 15 (2015)
Jang, M.-J., Löbrich, S.: Radial limits of the universal mock theta function \(g_3\). Proc. Amer. Math. Soc. 145(3), 925–935 (2017)
Kimport, S.: Quantum modular forms, mock modular forms, and partial theta functions. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Yale University (2015)
Kontsevich, M.: Lecture. Max Planck Institute for Mathematics, Bonn Germany (1997)
Lewis, J., Zagier, D.: Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis. Res. Math. Sci. 6(1), 24 (2019)
Lovejoy, J.: Quantum \(q\)-series identities. Hardy-Ramanujan J. 44, 61–73 (2021)
Lovejoy, J.: Bailey pairs and strange identities. J. Korean Math. Soc. 59(5), 1015–1045 (2022)
Lovejoy, J., Sarma, R: Bailey pairs, radial limits of \(q\)-hypergeometric false theta functions, and a conjecture of Hikami. arXiv:2402.11529, February 18 (2024)
Rolen, L., Schneider, R.P.: A vector-valued quantum modular form. Arch. Math. (Basel) 101(1), 43–52 (2013)
Watson, G.N.: The final problem: an account of the mock theta functions. J. London Math. Soc. 2(2), 55–80 (1936)
Zagier, D.: Vassiliev invariants and a strange identity related to the Dedekind eta-function. Topology 40(5), 945–960 (2001)
Zagier, D.: Ramanujan’s mock theta functions and their applications (after Zwegers and Ono-Bringmann). Number 326, Exp. No. 986, vii–viii, 143–164. Séminaire Bourbaki. Vol. 2007/2008 (2009)
Zagier, D.: Quantum modular forms. In Quanta of maths, volume 11 of Clay Math. Proc., pp. 659–675. Amer. Math. Soc., Providence, RI (2010)
Zwegers, S.P.: Mock \(\theta \)-functions and real analytic modular forms. In \(q\)-series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000), volume 291 of Contemp. Math., pp. 269–277. Amer. Math. Soc., Providence, RI (2001)
Zwegers, S.P.: Mock Theta Functions. Thesis (Ph.D.)–Utrecht University (2002)
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The authors are grateful for support from National Science Foundation RUI Grant DMS-2200728 (PI = first author). The authors thank the anonymous referee, and Jeremy Lovejoy, for their helpful comments on earlier drafts of this paper.
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Folsom, A., Metacarpa, D. Quantum q-series and mock theta functions. Res Math Sci 11, 41 (2024). https://doi.org/10.1007/s40687-024-00447-w
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DOI: https://doi.org/10.1007/s40687-024-00447-w
Keywords
- Mock theta functions
- Quantum modular form
- q-series
- q-hypergeometric series
- Basic hypergeometric series
- Quantum q-series