3rd order mock theta functions

introduction

• Ramanujan's 3rd order mock theta function is defined by \[f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\]
  • [1]http://www.research.att.com/~njas/sequences/A000025
  • http://www.research.att.com/~njas/sequences/b000025.txt

 

• 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 is given by \[f(q) = 1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}\sim \sqrt\frac{t}{2\pi}\exp(\frac{(2\pi)^2}{24t}-\frac{t}{24})+o(1)\]

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

 

 

harmonic weak Maass

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

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

 

articles

• good introduction is given in Andrews article 
  
  • Partitions : at the interface of q-series and modular forms
  • [2]section 5
• The Final Problem : An Account of the Mock Theta Functions
  
  • Watson, G. N. (1936),  J. London Math. Soc. 11: 55–80
• Some asymptotic formulae for the mock theta series of Ramanujan
  
  • Dragonette, Leila A. (1952), 
  • Transactions of the American Mathematical Society 72: 474–500
• On the theorems of Watson and Dragonette for Ramanujan's mock theta functions
  
  • Andrews, George E. (1966)
  • American Journal of Mathematics 88: 454–490

 

 

history

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

 

 

 

 

encyclopedia

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

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

articles

 

• 논문정리
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html[3]
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/