3rd order mock theta functions
imported>Pythagoras0님의 2013년 3월 18일 (월) 15:00 판
introduction
- Ramanujan's 3rd order mock theta function is defined by
\[f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\]
- 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
- 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
\[R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta_{1/2}(n^2y/6)q^{-n^2/24}\] 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
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
computational resources