"Rank of partition and mock theta conjecture"의 두 판 사이의 차이

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<h5>order 3 Ramanujan mock theta function</h5>
 
<h5>order 3 Ramanujan mock theta function</h5>
  
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* [[3rd order mock theta functions]]
 
* <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>
 
* <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>
 
*  coefficients<br> 1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244<br>[http://www.research.att.com/%7Enjas/sequences/A000025 http://www.research.att.com/~njas/sequences/A000025]<br>[http://www.research.att.com/%7Enjas/sequences/b000025.txt http://www.research.att.com/~njas/sequences/b000025.txt]<br>
 
*  coefficients<br> 1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244<br>[http://www.research.att.com/%7Enjas/sequences/A000025 http://www.research.att.com/~njas/sequences/A000025]<br>[http://www.research.att.com/%7Enjas/sequences/b000025.txt http://www.research.att.com/~njas/sequences/b000025.txt]<br>
 
* use mathematica '[[integer partitions]]' or the following
 
* use mathematica '[[integer partitions]]' or the following
  
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# Series[(1 + 4 Sum[(-1)^n q^(n (3 n + 1)/2)/(1 + q^n), {n, 1, 10}])/<br> Sum[(-1)^n q^(n (3 n + 1)/2), {n, -8, 8}], {q, 0, 100}]
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<h5>shadow</h5>
 
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* \Thate(24z)=q-5q^25+7q^{49}-11q^{121}+13q^{169}-\cdots
 
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* M_f(z)=q^{-1}f(q^24)+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz
 
  
 
 
 
 
119번째 줄: 111번째 줄:
 
<h5>articles</h5>
 
<h5>articles</h5>
  
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* [http://dx.doi.org/10.1007/s00222-005-0493-5 The f(q) mock theta function conjecture and partition ranks]<br>
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* [http://dx.doi.org/10.1007/s00222-005-0493-5 The f(q) mock theta function conjecture and partition ranks] Kathrin Bringmann and Ken Ono, Inventiones Mathematicae Volume 165, Number 2, 2006<br>
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** Kathrin Bringmann and Ken Ono, Inventiones Mathematicae Volume 165, Number 2, 2006
 
 
* [http://www.ingentaconnect.com/content/klu/rama/2003/00000007/F0030001/05142410 Partitions : at the interface of q-series and modular forms]<br>
 
* [http://www.ingentaconnect.com/content/klu/rama/2003/00000007/F0030001/05142410 Partitions : at the interface of q-series and modular forms]<br>
 
**  Andrews, George E., 2003<br>
 
**  Andrews, George E., 2003<br>

2011년 11월 11일 (금) 11:44 판

order 3 Ramanujan mock theta function
  1. Series[(1 + 4 Sum[(-1)^n q^(n (3 n + 1)/2)/(1 + q^n), {n, 1, 10}])/
    Sum[(-1)^n q^(n (3 n + 1)/2), {n, -8, 8}], {q, 0, 100}]

 

 

 

Andrews-Dragonette
  • [Dragonette1952] and [Andrews1966]
  • concerns the question of partitions with even rank and odd rank
  • rank of partition =  largest part - number of parts
    9의 분할인 {7,1,1}의 경우, rank=7-3=4
    9의 분할인 {4,3,1,1}의 경우, rank=4-4=0
  • \(N_e(n), N_o(n)\) number of partition with even rank and odd rank
  • \(p(n)=N_e(n)+N_o(n)\)
  • \(\alpha(n)=N_e(n)-N_o(n)\)
  • this is in fact the coefficient of mock theta function
    \(f(q) = \sum_{n\ge 0} \alpha(n)q^n\)
  • thus we need modularity of f(q) to get exact formula for \(\alpha(n)\) as \(p(n)\) was obtained by the circle method

 

 

harmonic Maass form of weight 1/2
  • Zweger's completion

 

 

construction of the Maass-Poincare series

 

 

generalization
  • crank

 

 

history

 

 

related items

 

 

books

 

encyclopedia

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

articles

 

 

experts on the field

 

 

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