"Rank of partition and mock theta conjecture"의 두 판 사이의 차이
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| − | <h5> | + | <h5>order 3 Ramanujan mock theta function</h5> |
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| + | * <math>f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;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><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> | ||
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| + | <h5>Andrews-Dragonette</h5> | ||
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| + | * rank of partition<br> 분할의 rank = 분할에서 가장 큰 수 - 분할의 크기<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>p(n)=N_e(n)+N_o(n)</math> | ||
| + | * <math>\alpha(n)=N_e(n)-N_o(n)</math> | ||
| + | * th | ||
| + | * <math>f(q) = \sum_{n\ge 0} \alpha(n)q^n</math> | ||
| + | * [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan]<br> | ||
| + | ** Dragonette, Leila A. (1952), | ||
| + | ** Transactions of the American Mathematical Society 72: 474–500 | ||
| + | * [http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions]<br> | ||
| + | ** Andrews, George E. (1966) | ||
| + | ** American Journal of Mathematics 88: 454–490 | ||
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| + | <h5>Maass-Poincare series</h5> | ||
| 83번째 줄: | 110번째 줄: | ||
<h5>articles</h5> | <h5>articles</h5> | ||
| − | + | * [http://www.maa.org/news/030807puzzlesolved.html Puzzle Solved: Ramanujan's Mock Theta Conjectures] | |
| − | + | * [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan]<br> | |
| − | * | + | ** Dragonette, Leila A. (1952), |
| + | ** Transactions of the American Mathematical Society 72: 474–500 | ||
| + | * [http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions]<br> | ||
| + | ** Andrews, George E. (1966) | ||
| + | ** American Journal of Mathematics 88: 454–490 | ||
| + | * [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.<br> | ||
| + | * [http://www.springerlink.com/content/5524655155350464/ The f(q) mock theta function conjecture and partition ranks]<br> | ||
| + | ** Inventiones Mathematicae, 2006 | ||
* [[2010년 books and articles|논문정리]] | * [[2010년 books and articles|논문정리]] | ||
* http://www.ams.org/mathscinet | * http://www.ams.org/mathscinet | ||
2010년 3월 3일 (수) 17:24 판
order 3 Ramanujan mock theta function
- \(f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} = {2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\)
http://www.research.att.com/~njas/sequences/A000025
http://www.research.att.com/~njas/sequences/b000025.txt
Andrews-Dragonette
- rank of partition
분할의 rank = 분할에서 가장 큰 수 - 분할의 크기
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
- \(p(n)=N_e(n)+N_o(n)\)
- \(\alpha(n)=N_e(n)-N_o(n)\)
- th
- \(f(q) = \sum_{n\ge 0} \alpha(n)q^n\)
- 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
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- Puzzle Solved: Ramanujan's Mock Theta Conjectures
- 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
- Partitions : at the interface of q-series and modular forms
- Andrews, George E.
- Andrews, George E.
- The f(q) mock theta function conjecture and partition ranks
- Inventiones Mathematicae, 2006
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