The odd moments of ranks and cranks

George E. Andrews, Song Heng Chan, Byungchan Kim

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)77-91
Number of pages15
JournalJournal of Combinatorial Theory. Series A
Volume120
Issue number1
DOIs
StatePublished - Jan 1 2013

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All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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