Examples of mock modular forms

mock theta function is essentially a holomorphic part of Mock modular forms of weight 1/2

\(f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} = {2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\)
[2]http://www.research.att.com/~njas/sequences/A000025
http://www.research.att.com/~njas/sequences/b000025.txt
good introduction is given in Andrews article
- Partitions : at the interface of q-series and modular forms
- [3]section 5
the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by of (Andrews 1966) and Dragonette (1952) converges to the coefficients (Bringmann & Ono 2006).
In particular Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper half-plane, that resemble those of modular forms of weight 1/2 with poles at the cusps.
The Final Problem : An Account of the Mock Theta Functions
- Watson, G. N. (1936), J. London Math. Soc. 11: 55–80
Some asymptotic formulae for the mock theta series of Ramanujan
- Dragonette, Leila A. (1952),
- Transactions of the American Mathematical Society 72: 474–500
On the theorems of Watson and Dragonette for Ramanujan's mock theta functions
- Andrews, George E. (1966)
- American Journal of Mathematics 88: 454–490

G.E. Andrews, The fifth and seventh order mock theta functions. Trans. Amer. Math. Soc. 293 (1986), pp. 113–134
Andrews, George E. (1988), "Ramanujan's fifth order mock theta functions as constant terms", Ramanujan revisited (Urbana-Champaign, Ill., 1987),
Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions
- Basil Gordon and Richard J. Mcintosh

Hickerson, Dean (1988), "On the seventh order mock theta functions", Inventiones Mathematicae94 (3): 661–677, doi:10.1007/BF01394280, MR 969247, ISSN 0020-9910

Choi, Youn-Seo (1999), "Tenth order mock theta functions in Ramanujan's lost notebook", Inventiones Mathematicae136 (3): 497–569, doi:10.1007/s002220050318, MR 1695205, ISSN 0020-9910
Choi, Youn-Seo (2000), "Tenth order mock theta functions in Ramanujan's lost notebook. II", Advances in Mathematics156 (2): 180–285, doi:10.1006/aima.2000.1948, MR 1808245, ISSN 0001-8708
Choi, Youn-Seo (2007), "Tenth order mock theta functions in Ramanujan's lost notebook. III", Proceedings of the London Mathematical Society. Third Series 94 (1): 26–52, doi:10.1112/plms/pdl006, MR 2293464, ISSN 0024-6115
Choi, Youn-Seo (2002), "Tenth order mock theta functions in Ramanujan's lost notebook. IV", Transactions of the American Mathematical Society354 (2): 705–733, doi:10.1090/S0002-9947-01-02861-6, MR 1862564, ISSN 0002-9947,
Zwegers' new proof http://mathsci.ucd.ie/~zwegers/papers/006.pdf

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