3rd order mock theta functions

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

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

  • 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.

asymptotics at 1

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 = 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

computational resources