TY - JOUR
T1 - The odd moments of ranks and cranks
AU - Andrews, George E.
AU - Chan, Song Heng
AU - Kim, Byungchan
N1 - Funding Information:
E-mail addresses: [email protected] (G.E. Andrews), [email protected] (S.H. Chan), [email protected] (B. Kim). 1 The author was supported by National Security Agency, NSA grant H98230-12-1-0205. 2 The author was partially supported by Nanyang Technological University Academic Research Fund, project number RG68/10. 3 The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF2011-0009199).
PY - 2013
Y1 - 2013
N2 - In this paper, we modify the standard definition of moments of ranks and cranks such that odd moments no longer trivially vanish. Denoting the new k-th rank (resp. crank) moments by N;k(n) (resp. M;k(n)), we prove the following inequality between the first rank and crank moments:. M;1(n)>N;1(n). This inequality motivates us to study a new counting function, ospt(n), which is equal to M;1(n)-N;1(n). We also discuss higher order moments of ranks and cranks. Surprisingly, for every higher order moments of ranks and cranks, the following inequality holds:. M;k(n)>N;k(n). This extends F.G. Garvan's result on the ordinary moments of ranks and cranks.
AB - In this paper, we modify the standard definition of moments of ranks and cranks such that odd moments no longer trivially vanish. Denoting the new k-th rank (resp. crank) moments by N;k(n) (resp. M;k(n)), we prove the following inequality between the first rank and crank moments:. M;1(n)>N;1(n). This inequality motivates us to study a new counting function, ospt(n), which is equal to M;1(n)-N;1(n). We also discuss higher order moments of ranks and cranks. Surprisingly, for every higher order moments of ranks and cranks, the following inequality holds:. M;k(n)>N;k(n). This extends F.G. Garvan's result on the ordinary moments of ranks and cranks.
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U2 - 10.1016/j.jcta.2012.07.001
DO - 10.1016/j.jcta.2012.07.001
M3 - Article
AN - SCOPUS:84863840305
SN - 0097-3165
VL - 120
SP - 77
EP - 91
JO - Journal of Combinatorial Theory. Series A
JF - Journal of Combinatorial Theory. Series A
IS - 1
ER -