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.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics