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://oeis.org/A000025

• 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 \[b(q)=q^{1/24}\frac{\eta(\tau)}{\eta(2\tau)}\theta(-q)\]
• let \(\zeta\) be even \(2k\) order root of unity

\[ \lim_{q\to \zeta} f(q)-(-1)^k b(q)=-4\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1} \]

• if \(k=2\), as \(q\to i\), \(f(q)-b(q)\to 4i\)

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

\[R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta(n^2y/6)q^{-n^2/24}\] 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 \[R_3(\tau)=(i/2)^{k-1} \int_{-\overline\tau}^{i\infty} (z+\tau)^{-k}\overline{g(-\overline z)}\,dz\]

where \(g\) is the shadow \[g_3(z)=\sum_{n\equiv 1\pmod 6}nq^{n^2/24}\]

shadow

• shadow = weight 3/2 theta function
• \(\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots\)
• \(M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz\)

related items

• Rank of partition and mock theta conjecture

computational resources

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

expositions

• Rolen, Ramanujan's mock theta functions.pdf
• Ono, https://docs.google.com/file/d/0B8XXo8Tve1cxTks3a095aGRqcGs/edit
• Mathematical proof reveals magic of Ramanujan's genius 2012-11-8
• Andrews, George E. 2003. “Partitions: At the Interface of Q-Series and Modular Forms.” The Ramanujan Journal 7 (1-3) (March 1): 385–400. doi:10.1023/A:1026224002193.
  • good introduction is given in section 5

articles

• Min-Joo Jang, Byungchan Kim, On spt-crank type functions, http://arxiv.org/abs/1603.05608v1
• George E. Andrews, Atul Dixit, Daniel Schultz, Ae Ja Yee, Overpartitions related to the mock theta function \(ω(q)\), http://arxiv.org/abs/1603.04352v1
• 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