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The crank moments weighted by the parity of cranks

Research paper by Kathy Q. Ji, Alice X. H. Zhao

Indexed on: 22 Oct '16Published on: 07 Oct '16Published in: The Ramanujan Journal



Abstract

Abstract In this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\) . When \(k=0\) , the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\) , where \(M_e(n)\) (resp. \(M_o(n)\) ) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\) .AbstractIn this note, we introduce the 2kth crank moment \(\mu _{2k}(-1,n)\) weighted by the parity of cranks and show that \((-1)^n \mu _{2k}(-1,n)>0\) for \(n\ge k \ge 0\) . When \(k=0\) , the inequality \((-1)^n \mu _{2k}(-1,n)>0\) reduces to Andrews and Lewis’s inequality \((-1)^n(M_e(n)-M_o(n))>0\) for \(n\ge 0\) , where \(M_e(n)\) (resp. \(M_o(n)\) ) denotes the number of partitions of n with even (resp. odd) crank. Several generating functions of \(\mu _{2k}(-1,n)\) are also studied in order to show the positivity of \((-1)^n\mu _{2k}(-1,n)\) .k \(\mu _{2k}(-1,n)\) \(\mu _{2k}(-1,n)\) \((-1)^n \mu _{2k}(-1,n)>0\) \((-1)^n \mu _{2k}(-1,n)>0\) \(n\ge k \ge 0\) \(n\ge k \ge 0\) \(k=0\) \(k=0\) \((-1)^n \mu _{2k}(-1,n)>0\) \((-1)^n \mu _{2k}(-1,n)>0\) \((-1)^n(M_e(n)-M_o(n))>0\) \((-1)^n(M_e(n)-M_o(n))>0\) \(n\ge 0\) \(n\ge 0\) \(M_e(n)\) \(M_e(n)\) \(M_o(n)\) \(M_o(n)\)n \(\mu _{2k}(-1,n)\) \(\mu _{2k}(-1,n)\) \((-1)^n\mu _{2k}(-1,n)\) \((-1)^n\mu _{2k}(-1,n)\)