# The crank moments weighted by the parity of cranks

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

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

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