# 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 $$\mu _{2k}(-1,n)$$ $$\mu _{2k}(-1,n)$$ $$(-1)^n\mu _{2k}(-1,n)$$ $$(-1)^n\mu _{2k}(-1,n)$$