The odd moments of ranks and cranks

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. Garvanʼs result on the ordinary moments of ranks and cranks.