"Rank of partition and mock theta conjecture"의 두 판 사이의 차이
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| 2번째 줄: | 2번째 줄: | ||
* [[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> | * [[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> | ||
| − | * coefficients | + | * 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 |
| 10번째 줄: | 10번째 줄: | ||
* '''[Dragonette1952]''' and '''[Andrews1966]''' | * '''[Dragonette1952]''' and '''[Andrews1966]''' | ||
* concerns the question of partitions with even rank and odd rank | * concerns the question of partitions with even rank and odd rank | ||
| − | * rank of partition = largest part - number of parts | + | * 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 |
* <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 the [[3rd order mock theta functions]] | * this is in fact the coefficient of the [[3rd order mock theta functions]] | ||
| − | :<math>f(q) = \sum_{n\ge 0} \alpha(n)q^n</math | + | :<math>f(q) = \sum_{n\ge 0} \alpha(n)q^n</math> |
* 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 | * 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 | ||
| 71번째 줄: | 71번째 줄: | ||
* [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 | * [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 | ||
| − | * George E. Andrews and F. G. Garvan, [http://dx.doi.org/10.1016/0001-8708%2889%2990070-4 Ramanujan's “Lost” Notebook VI: The mock theta conjectures] 1989 | + | * George E. Andrews and F. G. Garvan, [http://dx.doi.org/10.1016/0001-8708%2889%2990070-4 Ramanujan's “Lost” Notebook VI: The mock theta conjectures] 1989 |
* Hickerson, Dean, <cite class="" id="CITEREFHickerson1988" style="line-height: 2em; font-style: normal;">A proof of the mock theta conjectures</cite> (1988), <cite style="line-height: 2em; font-style: normal;"><em style="line-height: 2em;">[http://en.wikipedia.org/wiki/Inventiones_Mathematicae Inventiones Mathematicae]</em> '''94''' (3): 639–660, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1007%2FBF01394279 10.1007/BF01394279], [http://en.wikipedia.org/wiki/Mathematical_Reviews MR][http://www.ams.org/mathscinet-getitem?mr=969247 969247], [http://en.wikipedia.org/wiki/International_Standard_Serial_Number ISSN] [http://worldcat.org/issn/0020-9910 0020-9910]</cite> | * Hickerson, Dean, <cite class="" id="CITEREFHickerson1988" style="line-height: 2em; font-style: normal;">A proof of the mock theta conjectures</cite> (1988), <cite style="line-height: 2em; font-style: normal;"><em style="line-height: 2em;">[http://en.wikipedia.org/wiki/Inventiones_Mathematicae Inventiones Mathematicae]</em> '''94''' (3): 639–660, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1007%2FBF01394279 10.1007/BF01394279], [http://en.wikipedia.org/wiki/Mathematical_Reviews MR][http://www.ams.org/mathscinet-getitem?mr=969247 969247], [http://en.wikipedia.org/wiki/International_Standard_Serial_Number ISSN] [http://worldcat.org/issn/0020-9910 0020-9910]</cite> | ||
| − | * '''[Andrews1966]'''[http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions] | + | * '''[Andrews1966]'''[http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions] |
** Andrews, George E. (1966), American Journal of Mathematics 88: 454–490 | ** Andrews, George E. (1966), American Journal of Mathematics 88: 454–490 | ||
| − | * '''[Dragonette1952]'''[http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan] | + | * '''[Dragonette1952]'''[http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan] |
** Dragonette, Leila A. (1952), Transactions of the American Mathematical Society 72: 474–500 | ** Dragonette, Leila A. (1952), Transactions of the American Mathematical Society 72: 474–500 | ||
* <cite class="" id="CITEREFWatson1937" style="line-height: 2em; font-style: normal;">Watson, G. N. (1937), "The Mock Theta Functions (2)", <em style="line-height: 2em;">Proc. London Math. Soc.</em> '''s2-42''': 274–304, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1112%2Fplms%2Fs2-42.1.274 10.1112/plms/s2-42.1.274]</cite> | * <cite class="" id="CITEREFWatson1937" style="line-height: 2em; font-style: normal;">Watson, G. N. (1937), "The Mock Theta Functions (2)", <em style="line-height: 2em;">Proc. London Math. Soc.</em> '''s2-42''': 274–304, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1112%2Fplms%2Fs2-42.1.274 10.1112/plms/s2-42.1.274]</cite> | ||
2020년 11월 16일 (월) 03:36 판
order 3 Ramanujan mock theta function
- 3rd order mock theta functions \[f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\]
- 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
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 the 3rd order mock theta functions
\[f(q) = \sum_{n\ge 0} \alpha(n)q^n\]
- 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
computational resource
expositions
- Puzzle Solved: Ramanujan's Mock Theta Conjectures
- Partitions : at the interface of q-series and modular forms Andrews, George E., 2003
articles
- The f(q) mock theta function conjecture and partition ranks Kathrin Bringmann and Ken Ono, Inventiones Mathematicae Volume 165, Number 2, 2006
- George E. Andrews and F. G. Garvan, Ramanujan's “Lost” Notebook VI: The mock theta conjectures 1989
- Hickerson, Dean, A proof of the mock theta conjectures (1988), Inventiones Mathematicae 94 (3): 639–660, doi:10.1007/BF01394279, MR969247, ISSN 0020-9910
- [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
- [Dragonette1952]Some asymptotic formulae for the mock theta series of Ramanujan
- Dragonette, Leila A. (1952), Transactions of the American Mathematical Society 72: 474–500
- Watson, G. N. (1937), "The Mock Theta Functions (2)", Proc. London Math. Soc. s2-42: 274–304, doi:10.1112/plms/s2-42.1.274
- Watson, G. N. (1936), "The Final Problem : An Account of the Mock Theta Functions", J. London Math. Soc. 11: 55–80, doi:10.1112/jlms/s1-11.1.55