MOCK THETA FUNCTIONS AND QUANTUM MODULAR FORMS

被引：55

作者：

Folsom, Amanda ^{[1
]}

Ono, Ken ^{[2
]}

Rhoades, Robert C. ^{[3
]}

机构：

[1] Yale Univ, Dept Math, New Haven, CT 06520 USA

[2] Emory Univ, Dept Math, Atlanta, GA 30322 USA

[3] Stanford Univ, Dept Math, Stanford, CA 94305 USA

来源：

FORUM OF MATHEMATICS PI | 2013年 / 1卷

关键词：

D O I：

10.1017/fmp.2013.3

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

Ramanujan's last letter to Hardy concerns the asymptotic properties of modular forms and his 'mock theta functions'. For the mock theta function f(q), Ramanujan claims that as q approaches an even-order 2k root of unity, we have f(q) - (-1)(k) (1-q) (1 - q(3))(1 - q(5)) ... (1 - 2q + 2q(4) - ... ) =O(1). We prove Ramanujan's claim as a special case of a more general result. The implied constants in Ramanujan's claim are not mysterious. They arise in Zagier's theory of 'quantum modular forms'. We provide explicit closed expressions for these 'radial limits' as values of a 'quantum' q-hypergeometric function which underlies a new relationship between Dyson's rank mock theta function and the Andrews-Garvan crank modular form. Along these lines, we show that the Rogers-Fine false theta-functions, functions which have not been well understood within the theory of modular forms, specialize to quantum modular forms.

引用

页数：27

共 45 条

[1]

Andrews G. E., 2013, ALGEBRA NUM IN PRESS

[2]

Andrews G. E., 2009, RAMANUFANS LOST NO 2

[3]

Andrews G. E., RAMANUJAN J IN PRESS

[4] ON THEOREMS OF WATSON AND DRAGONETTE FOR RAMANUJANS MOCK THETA FUNCTIONS [J].