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

2012년 10월 29일 (월) 10:55 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

[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

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