Abstract
Kang discovered a formula expressing the universal mock \(\theta \)-function \(g_{{{\scriptstyle 2}}}\) in terms of Zwegers’ \(\mu \)-function and a \(\theta \)-quotient. By modifying the elliptic variables in \(\mu \) the \(\theta \)-quotient can be removed from Kang’s formula. We also obtain a formula expressing \(\mu \) in terms of \(g_{{{\scriptstyle 2}}}\), proving that \(\mu \) is not more general than \(g_{{{\scriptstyle 2}}}\), even though it has one more elliptic variable.
Dedicated to Krishnaswami Alladi on the occasion of his 60th birthday
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
P. Appell, Sur les fonctions doublement périodiques de troisième espèce. Annales Scientifiques de l’École Normale Supérieure 1, 135–164 (1884)
Y.-S. Choi, Tenth order mock theta functions in Ramanujan’s lost notebook. Invent. Math. 136, 497–569 (1999)
F.G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11. Trans. AMS 305, 47–77 (1988)
B. Gordon, R.J. McIntosh, A survey of classical mock theta functions, Partitions, q-Series, and Modular Forms, vol. 23, Developments in Mathematics (Springer, New York, 2012), pp. 95–144
D. Hickerson, A proof of the mock theta conjectures. Invent. Math. 94, 639–660 (1988)
D. Hickerson, E. Mortenson, Hecke-type double sums, Appell–Lerch sums, and mock theta functions, I. Proc. Lond. Math. Soc. 109, 382–422 (2014)
S.-Y. Kang, Mock Jacobi forms in basic hypergeometric series. Compositio Mathematica 145, 553–565 (2009)
M. Lerch, Bemerkungen zur Theorie der elliptischen Funktionen. Jahrbuch über die Fortschritte der Mathematik 24, 442–445 (1892)
R.J. McIntosh, Some identities for Appell–Lerch sums and a universal mock theta function, to appear in Ramanujan J
S.P. Zwegers, Mock \(\vartheta \)-functions and real analytic modular forms, in \(q\)-Series with Applications to Combinatorics, Number Theory, and Physics, vol. 291, Contemporary Mathematics, ed. by B.C. Berndt, K. Ono (2001), pp. 269–277
S.P. Zwegers, Mock Theta Functions, Ph.D. thesis, Universiteit Utrecht, 2002
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
McIntosh, R.J. (2017). On the Universal Mock Theta Function \(g_{{{\scriptstyle 2}}}\) and Zwegers’ \(\mu \)-Function. In: Andrews, G., Garvan, F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham. https://doi.org/10.1007/978-3-319-68376-8_28
Download citation
DOI: https://doi.org/10.1007/978-3-319-68376-8_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68375-1
Online ISBN: 978-3-319-68376-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)