The crank moments weighted by the parity of cranks

On a generalized crank for k-colored partitions

“…The proofs of (3.4) and (3.5) are similar to those of the corresponding results in [20], thus we omit the details here. Now we are ready to derive (3.2) and (3.3).…”

“…Theorem 1.2 in[20]). For n ≥ j ≥ 0, (−1) n µ 2j (−1, n) > 0.Motivated by the work of Ji and Zhao, we consider the 2j th symmetrized moments of k-colored partitions weighted by the parity of k-cranks, defined asµ 2j,k (−1, n) := n m=−n m + j − 1 2j (−1) m M k (m, n), for k ≥ 2.…”

Congruences modulo 9 for bipartitions with designated summands

“…As a natural analog to the 2k-th crank moments µ 2k (−1, n) weighted by the parity of cranks due to Ji and Zhao [13], we define the 2k-th pd-crank moments of bipartitions with designated summands weighted by the parity of pd-cranks as given by…”

“…With the aid of the proof of Theorem 1.3 of Ji and Zhao [13] and applying Andrews' j-fold generalization of q-Whipple's theorem [1], we obtain the generating function of µ 2k,bd (−1, n) as…”

“…Since the fact that M(m, n) = M(−m, n) [2, (1.9)], it is easy to see that µ 2k+1 (n) = 0. In a recent work, Ji and Zhao [13] introduced the crank moments weighted by the parity of cranks as given by…”