Examples of mock modular forms

Examples

 

classify which examples are of which use

 

 

mock theta function

• mock theta function is essentially a mock modular form of weight 1/2

 

mock theta conjectures

• Watson, G. N. (1936), "The Final Problem : An Account of the Mock Theta Functions", J. London Math. Soc.11: 55–80, doi:10.1112/jlms/s1-11.1.55
• Watson, G. N. (1937), "The Mock Theta Functions (2)", Proc. London Math. Soc.s2-42: 274–304, doi:10.1112/plms/s2-42.1.274
• Ramanujan's “Lost” Notebook VI: The mock theta conjectures
  
  • George E. Andrews and F. G. Garvan   
• Hickerson, Dean (1988), "A proof of the mock theta conjectures", Inventiones Mathematicae94 (3): 639–660, doi:10.1007/BF01394279, MR969247, ISSN0020-9910
  [1]

 

order 3

• \(f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} = {2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\)
• 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.
• Some asymptotic formulae for the mock theta series of Ramanujan
  
  • Dragonette, Leila A. (1952), 
  • Transactions of the American Mathematical Society 72: 474–500
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order 5

• 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

 

order 7

• Hickerson, Dean (1988), "On the seventh order mock theta functions", Inventiones Mathematicae94 (3): 661–677, doi:10.1007/BF01394280, MR969247, ISSN0020-9910

 

 

order 10

• Choi, Youn-Seo (1999), "Tenth order mock theta functions in Ramanujan's lost notebook", Inventiones Mathematicae136 (3): 497–569, doi:10.1007/s002220050318, MR1695205, ISSN0020-9910
• 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, MR1808245, ISSN0001-8708
• 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, MR1862564, ISSN0002-9947, http://www.jstor.org/stable/2693766
• Choi, Youn-Seo (2007), "Tenth order mock theta functions in Ramanujan's lost notebook. III", Proceedings of the London Mathematical Society. Third Series94 (1): 26–52, doi:10.1112/plms/pdl006, MR2293464, ISSN0024-6115

http://mathsci.ucd.ie/~zwegers/papers/006.pdf

 

 

 

 

 

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관련된 다른 주제들

• Mock modular forms

 

표준적인 도서 및 추천도서

• Ramanujan's Lost Notebook: Part I
• Ramanujan's Lost Notebook: Part II
• 찾아볼 수학책
• http://gigapedia.info/1/Polyogarithms
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