"3rd order mock theta functions"의 두 판 사이의 차이

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==introduction==
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* Ramanujan's 3rd order mock theta function is defined by
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:<math>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}</math>
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** http://oeis.org/A000025
  
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* 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 ([http://www.springerlink.com/content/5524655155350464/ Bringmann & Ono 2006]).
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** see [[Rank of partition and mock theta conjecture]]
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* 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.
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==asymptotics at 1==
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* If <math>q=e^{-t}</math>, around <math>t\sim 0</math>, the asymptotic behavior is given by :<math>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</math>
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* see also [[Asymptotic analysis of basic hypergeometric series]]
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==asymptotic behavior at roots of unity==
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* the series converges for <math>|q|<1</math> and the roots of unity <math>q</math> at odd order
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* For even order roots of unity, <math>f(q)</math> has exponential singularities but there is a nice result to describe this behavior
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* let us define :<math>b(q)=(1-q)(1-q^3)(1-q^5)\cdots (1-2q+2q^4-\cdots)</math>, or we can write it as :<math>b(q)=q^{1/24}\frac{\eta(\tau)}{\eta(2\tau)}\theta(-q)</math>
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* let <math>\zeta</math> be even <math>2k</math> order root of unity
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:<math>
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\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}
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</math>
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* if <math>k=2</math>, as <math>q\to i</math>, <math>f(q)-b(q)\to 4i</math>
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==harmonic weak Maass form==
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* We have a weight k=1/2, harmonic weak Maass form <math>h_3</math> under <math>\Gamma(2)</math> defined by :<math>h_3(\tau)=q^{-1/24}f(q)+R_3(q)</math> where
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:<math>R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta(n^2y/6)q^{-n^2/24}</math> where
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:<math>\displaystyle \beta(t) = \int_t^\infty u^{-1/2} e^{-\pi u} \,du=2\int_{\sqrt{x}}^{\infty} e^{-\pi t^2}\,dt</math>
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* Note that this can be rewritten as :<math>R_3(\tau)=(i/2)^{k-1} \int_{-\overline\tau}^{i\infty} (z+\tau)^{-k}\overline{g(-\overline z)}\,dz</math>
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where <math>g</math> is the shadow
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:<math>g_3(z)=\sum_{n\equiv 1\pmod 6}nq^{n^2/24}</math>
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===shadow===
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* shadow = weight 3/2 theta function
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* <math>\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots</math>
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* <math>M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz</math>
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==related items==
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* [[Rank of partition and mock theta conjecture]]
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==computational resources==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit
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==expositions==
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* [https://docs.google.com/file/d/0B8XXo8Tve1cxOFZTUldUc1l1a2s/edit?usp=drivesdk Rolen, Ramanujan's mock theta functions.pdf]
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* Ono, https://docs.google.com/file/d/0B8XXo8Tve1cxTks3a095aGRqcGs/edit
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* [http://www.newscientist.com/article/mg21628904.200-mathematical-proof-reveals-magic-of-ramanujans-genius.html Mathematical proof reveals magic of Ramanujan's genius] 2012-11-8
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* 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.
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** good introduction is given in  section 5
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==articles==
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* Min-Joo Jang, Byungchan Kim, On spt-crank type functions, http://arxiv.org/abs/1603.05608v1
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* George E. Andrews, Atul Dixit, Daniel Schultz, Ae Ja Yee, Overpartitions related to the mock theta function <math>ω(q)</math>, http://arxiv.org/abs/1603.04352v1
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* Watson, G. N. [http://dx.doi.org/10.1112%2Fjlms%2Fs1-11.1.55 The Final Problem : An Account of the Mock Theta Functions](1936),  J. London Math. Soc. 11: 55–80
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* Dragonette, Leila A. [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan](1952), Transactions of the American Mathematical Society 72: 474–500
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* Andrews, George E.  [http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions](1966) American Journal of Mathematics 88: 454–490
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[[분류:개인노트]]
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[[분류:math and physics]]
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[[분류:mock modular forms]]
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[[분류:math]]
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[[분류:migrate]]

2020년 12월 28일 (월) 05:05 기준 최신판

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


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


computational resources


expositions



articles

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