3rd order mock theta functions

introduction

• Ramanujan's 3rd order mock theta function is defined by

\[f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\]

• • http://www.research.att.com/~njas/sequences/A000025
  • http://www.research.att.com/~njas/sequences/b000025

 

• 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).
  • see Rank of partition and mock theta conjecture
• 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.

asymptotics at 1

• If \(q=e^{-t}\), around \(t\sim 0\), the asymptotic behavior is given by \[f(q) = 1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}\sim 4/3\]
• see also Asymptotic analysis of basic hypergeometric series

asymptotic behavior at roots of unity

• the series converges for $|q|<1$ and the roots of unity $q$ at odd order
• For even order roots of unity, $f(q)$ has exponential singularities but there is a nice result to describe this behavior
• let us define $$b(q)=(1-q)(1-q^3)(1-q^5)\cdots (1-2q+2q^4-\cdots)$$, or we can write it as $$b(q)=q^{1/24}\frac{\eta(\tau)}{\eta(2\tau)}\theta(-q)$$

 

harmonic weak Maass form

• We have a weight k=1/2, harmonic weak Maass form $h_3$ under \(\Gamma(2)\) defined by \[h_3(\tau)=q^{-1/24}f(q)+R_3(q)\] where

\(R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta_{1/2}(n^2y/6)q^{-n^2/24}\) where \(\displaystyle \beta(t) = \int_t^\infty u^{-1/2} e^{-\pi u} \,du=2\int_{\sqrt{x}}^{\infty} e^{-\pi t^2}\,dt\)

• Note that this can be rewritten as \[R_3(\tau)=(i/2)^{k-1} \int_{-\overline\tau}^{i\infty} (z+\tau)^{-k}\overline{g(-\overline z)}\,dz\]
• shadow = weight 3/2 theta function \[g_3(z)=\sum_{n\equiv 1\pmod 6}nq^{n^2/24}\]

 

shadow

• \(\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots\)
• \(M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz\)

 

expositions

• Mathematical proof reveals magic of Ramanujan's genius 2012-11-8
• good introduction is given in Andrews article
  • Partitions : at the interface of q-series and modular forms
  • section 5

articles

• Watson, G. N. The Final Problem : An Account of the Mock Theta Functions(1936),  J. London Math. Soc. 11: 55–80
• Dragonette, Leila A. Some asymptotic formulae for the mock theta series of Ramanujan(1952), Transactions of the American Mathematical Society 72: 474–500
• Andrews, George E. On the theorems of Watson and Dragonette for Ramanujan's mock theta functions(1966) American Journal of Mathematics 88: 454–490

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

 

computational resources

• https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/