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

2012년 11월 3일 (토) 08:27 판

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
http://www.research.att.com/~njas/sequences/A000025
http://www.research.att.com/~njas/sequences/b000025.txt
use mathematica 'integer partitions' or the following

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}]

[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

The f(q) mock theta function conjecture and partition ranks Kathrin Bringmann and Ken Ono, Inventiones Mathematicae Volume 165, Number 2, 2006
[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

1988 Hickerson

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
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
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, MR 969247, ISSN 0020-9910

==TeX ==

@@ 107번째 줄: / 107번째 줄: @@
+==expositions==
+*  [http://link.springer.com/article/10.1023%2FA%3A1026224002193?LI=true Partitions : at the interface of q-series and modular forms] Andrews, George E., 2003<br>
 ==articles==
 * [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>
-* [http://www.ingentaconnect.com/content/klu/rama/2003/00000007/F0030001/05142410 Partitions : at the interface of q-series and modular forms] Andrews, George E., 2003<br>
 * '''[Dragonette1952]'''[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