"3rd order mock theta functions"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
+ | * Ramanujan's 3rd order mock theta function is defined by | ||
+ | :<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> | ||
+ | ** 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]). | ||
+ | ** 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. | ||
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− | * <math>f(q) = \sum_{n\ge | + | ==asymptotics at 1== |
+ | * 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> | ||
+ | * see also [[Asymptotic analysis of basic hypergeometric series]] | ||
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==asymptotic behavior at roots of unity== | ==asymptotic behavior at roots of unity== | ||
− | * the series converges for | + | * the series converges for <math>|q|<1</math> and the roots of unity <math>q</math> at odd order |
− | * For even order roots of unity, | + | * For even order roots of unity, <math>f(q)</math> has exponential singularities but there is a nice result to describe this behavior |
− | + | * 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> | |
− | + | * let <math>\zeta</math> be even <math>2k</math> order root of unity | |
+ | :<math> | ||
+ | \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} | ||
+ | </math> | ||
+ | * if <math>k=2</math>, as <math>q\to i</math>, <math>f(q)-b(q)\to 4i</math> | ||
+ | |||
+ | ==harmonic weak Maass form== | ||
+ | * 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 | ||
+ | :<math>R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta(n^2y/6)q^{-n^2/24}</math> where | ||
+ | :<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> | ||
+ | * 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> | ||
+ | where <math>g</math> is the shadow | ||
+ | :<math>g_3(z)=\sum_{n\equiv 1\pmod 6}nq^{n^2/24}</math> | ||
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+ | ===shadow=== | ||
+ | * shadow = weight 3/2 theta function | ||
* <math>\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots</math> | * <math>\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots</math> | ||
* <math>M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz</math> | * <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== | ==related items== | ||
+ | * [[Rank of partition and mock theta conjecture]] | ||
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+ | ==computational resources== | ||
+ | * https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit | ||
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− | * | + | ==expositions== |
− | * http:// | + | * [https://docs.google.com/file/d/0B8XXo8Tve1cxOFZTUldUc1l1a2s/edit?usp=drivesdk Rolen, Ramanujan's mock theta functions.pdf] |
+ | * Ono, https://docs.google.com/file/d/0B8XXo8Tve1cxTks3a095aGRqcGs/edit | ||
+ | * [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 | ||
+ | * 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 | ||
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==articles== | ==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 <math>ω(q)</math>, http://arxiv.org/abs/1603.04352v1 | ||
+ | * 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 | ||
+ | * 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 | ||
+ | * 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]] | [[분류:math and physics]] | ||
[[분류:mock modular forms]] | [[분류:mock modular forms]] | ||
+ | [[분류:math]] | ||
+ | [[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
- 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
- \(\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots\)
- \(M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz\)
computational resources
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