Higher depth mock theta functions and q-hypergeometric series
-
Joshua Males
,
Andreas Mono
Abstract
In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.
Acknowledgements
We would like to thank Jeremy Lovejoy for insightful discussions on partition-theoretic aspects of this paper. In addition, we would like to thank Kathrin Bringmann for useful comments on an earlier version of this paper. We also thank the referee for many helpful suggestions.
References
[1] S. Alexandrov, S. Banerjee, J. Manschot and B. Pioline, Indefinite theta series and generalized error functions, Selecta Math. (N. S.) 24 (2018), no. 5, 3927–3972. 10.1007/s00029-018-0444-9Suche in Google Scholar
[2] S. Alexandrov and B. Pioline, Black holes and higher depth mock modular forms, Comm. Math. Phys. 374 (2020), no. 2, 549–625. 10.1007/s00220-019-03609-ySuche in Google Scholar
[3] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook. Part II, Springer, New York, 2009. Suche in Google Scholar
[4] K. Bringmann, Asymptotics for rank partition functions, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3483–3500. 10.1090/S0002-9947-09-04553-XSuche in Google Scholar
[5] K. Bringmann, A. Folsom, K. Ono and L. Rolen, Harmonic Maass Forms and Mock Modular Forms: Theory and Applications, Amer. Math. Soc. Colloq. Publ. 64, American Mathematical Society, Providence, 2017. 10.1090/coll/064Suche in Google Scholar
[6] K. Bringmann, A. Folsom and R. C. Rhoades, Partial theta functions and mock modular forms as q-hypergeometric series, Ramanujan J. 29 (2012), no. 1–3, 295–310. 10.1007/s11139-012-9370-1Suche in Google Scholar
[7]
K. Bringmann, J. Kaszian and A. Milas,
Higher depth quantum modular forms, multiple Eichler integrals, and
[8] K. Bringmann, J. Kaszian and A. Milas, Vector-valued higher depth quantum modular forms and higher Mordell integrals, J. Math. Anal. Appl. 480 (2019), no. 2, Article ID 123397. 10.1016/j.jmaa.2019.123397Suche in Google Scholar
[9] K. Bringmann, J. Kaszián and L. Rolen, Indefinite theta functions arising in Gromov–Witten theory of elliptic orbifolds, Camb. J. Math. 6 (2018), no. 1, 25–57. 10.4310/CJM.2018.v6.n1.a2Suche in Google Scholar
[10]
K. Bringmann and K. Ono,
The
[11] K. Bringmann and K. Ono, Dyson’s ranks and Maass forms, Ann. of Math. (2) 171 (2010), no. 1, 419–449. 10.4007/annals.2010.171.419Suche in Google Scholar
[12] J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90. 10.1215/S0012-7094-04-12513-8Suche in Google Scholar
[13] F. Calegari, Bloch groups, algebraic K-theory, units, and Nahm’s conjecture, preprint (2017), https://arxiv.org/abs/1712.04887. Suche in Google Scholar
[14] Y.-S. Choi, The basic bilateral hypergeometric series and the mock theta functions, Ramanujan J. 24 (2011), no. 3, 345–386. 10.1007/s11139-010-9269-7Suche in Google Scholar
[15] A. Dabholkar, S. Murthy and D. Zagier, Quantum black holes, wall crossing, and mock modular forms, preprint (2012), https://arxiv.org/abs/1208.4074. Suche in Google Scholar
[16] N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988. 10.1090/surv/027Suche in Google Scholar
[17] A. Folsom, Mock and mixed mock modular forms in the lower half-plane, Arch. Math. (Basel) 107 (2016), no. 5, 487–498. 10.1007/s00013-016-0951-xSuche in Google Scholar
[18] S. Garoufalidis and T. T. Q. Lê, Nahm sums, stability and the colored Jones polynomial, Res. Math. Sci. 2 (2015), Article ID 1. 10.1186/2197-9847-2-1Suche in Google Scholar
[19] B. Gordon and R. J. McIntosh, A Survey of Classical Mock Theta Functions, Dev. Math. 23, Springer, New York, 2012. 10.1007/978-1-4614-0028-8_9Suche in Google Scholar
[20] J. Lovejoy and R. Osburn, q-hypergeometric double sums as mock theta functions, Pacific J. Math. 264 (2013), no. 1, 151–162. 10.2140/pjm.2013.264.151Suche in Google Scholar
[21] J. Lovejoy and R. Osburn, Mock theta double sums, Glasg. Math. J. 59 (2017), no. 2, 323–348. 10.1017/S0017089516000197Suche in Google Scholar
[22] J. Males, A family of vector-valued quantum modular forms of depth two, Int. J. Number Theory 16 (2020), no. 1, 29–64. 10.1142/S1793042120500025Suche in Google Scholar
[23] C. Nazaroglu, r-tuple error functions and indefinite theta series of higher-depth, Commun. Number Theory Phys. 12 (2018), no. 3, 581–608. 10.4310/CNTP.2018.v12.n3.a4Suche in Google Scholar
[24] H. M. Srivastava, Some formulas of Srinivasa Ramanujan involving products of hypergeometric functions, Indian J. Math. 29 (1987), no. 1, 91–100. Suche in Google Scholar
[25] M. Vlasenko and S. Zwegers, Nahm’s conjecture: Asymptotic computations and counterexamples, Commun. Number Theory Phys. 5 (2011), no. 3, 617–642. 10.4310/CNTP.2011.v5.n3.a2Suche in Google Scholar
[26]
M. Westerholt-Raum,
[27] D. Zagier, The dilogarithm function, Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin (2007), 3–65. 10.1007/978-3-540-30308-4_1Suche in Google Scholar
[28] D. Zagier, Ramanujan’s mock theta functions and their applications (after Zwegers and Ono–Bringmann), Séminaire Bourbaki. Vol. 2007/2008. Exposés 982-996, Astérisque 326, Société Mathématique de France, Paris (2010), 143–146, Exp. No. 986. Suche in Google Scholar
[29] D. Zagier, Quantum modular forms, Quanta of Maths, Clay Math. Proc. 11, American Mathematical Society, Providence (2010), 659–675. Suche in Google Scholar
[30] S. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 2002. Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Higher depth mock theta functions and q-hypergeometric series
- Topological and algebraic properties of universal groups for right-angled buildings
- On the socles of characteristically inert subgroups of Abelian p-groups
- Priestley duality for MV-algebras and beyond
- The cardinality of μM,D‐orthogonal exponentials for the planar four digits
- Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
- Ordered fields dense in their real closure and definable convex valuations
- Third Hankel determinants for two classes of analytic functions with real coefficients
- A common range problem for model spaces
- Generalized Ricci flow on nilpotent Lie groups
- Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
- The role of the algebraic structure in Wold-type decomposition
- Incidences between Euclidean spaces over finite fields
- Cancellation in algebraic twisted sums on GL_m
Artikel in diesem Heft
- Frontmatter
- Higher depth mock theta functions and q-hypergeometric series
- Topological and algebraic properties of universal groups for right-angled buildings
- On the socles of characteristically inert subgroups of Abelian p-groups
- Priestley duality for MV-algebras and beyond
- The cardinality of μM,D‐orthogonal exponentials for the planar four digits
- Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
- Ordered fields dense in their real closure and definable convex valuations
- Third Hankel determinants for two classes of analytic functions with real coefficients
- A common range problem for model spaces
- Generalized Ricci flow on nilpotent Lie groups
- Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
- The role of the algebraic structure in Wold-type decomposition
- Incidences between Euclidean spaces over finite fields
- Cancellation in algebraic twisted sums on GL_m
