"Rank of partition and mock theta conjecture"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
<h5>order 3 Ramanujan mock theta function</h5> | <h5>order 3 Ramanujan mock theta function</h5> | ||
| − | * <math>f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} {q^{n^2}\ | + | * <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> | ||
| 10번째 줄: | 11번째 줄: | ||
* '''[Dragonette1952]''' and '''[Andrews1966]''' | * '''[Dragonette1952]''' and '''[Andrews1966]''' | ||
| − | * rank of partition | + | * concerns the question of partitions with even rank and odd rank |
| + | * rank of partition = largest part - number of parts<br> 9의 분할인 {7,1,1}의 경우, rank=7-3=4<br> 9의 분할인 {4,3,1,1}의 경우, rank=4-4=0<br> | ||
* <math>N_e(n), N_o(n)</math> number of partition with even rank and odd rank | * <math>N_e(n), N_o(n)</math> number of partition with even rank and odd rank | ||
* <math>p(n)=N_e(n)+N_o(n)</math> | * <math>p(n)=N_e(n)+N_o(n)</math> | ||
* <math>\alpha(n)=N_e(n)-N_o(n)</math> | * <math>\alpha(n)=N_e(n)-N_o(n)</math> | ||
* this is in fact the coefficient of mock theta function<br><math>f(q) = \sum_{n\ge 0} \alpha(n)q^n</math><br> | * this is in fact the coefficient of mock theta function<br><math>f(q) = \sum_{n\ge 0} \alpha(n)q^n</math><br> | ||
| − | * thus we need modularity of f(q) to get exact formula for <math>\alpha(n)</math> as <math>p(n)</math> obtained by the circle method | + | * thus we need modularity of f(q) to get exact formula for <math>\alpha(n)</math> as <math>p(n)</math> was obtained by the circle method |
| 37번째 줄: | 39번째 줄: | ||
<h5>generalization</h5> | <h5>generalization</h5> | ||
| − | + | * crank | |
2010년 3월 3일 (수) 18:40 판
order 3 Ramanujan mock theta function
- \(f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\)
- coefficients
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
http://www.research.att.com/~njas/sequences/A000025
http://www.research.att.com/~njas/sequences/b000025.txt
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
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articles[1]
- The f(q) mock theta function conjecture and partition ranks
- Inventiones Mathematicae, 2006
- Partitions : at the interface of q-series and modular forms
- Andrews, George E., 2003
- Andrews, George E., 2003
- [Dragonette1952]Some asymptotic formulae for the mock theta series of Ramanujan
- Dragonette, Leila A. (1952), Transactions of the American Mathematical Society 72: 474–500
-
[Andrews1966]On the theorems of Watson and Dragonette for Ramanujan's mock theta functions
- Andrews, George E. (1966), American Journal of Mathematics 88: 454–490
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