  In this paper, we extend recent results of Assaf and McNamara on skew Pieri
rule and skew Murnaghan-Nakayama rule to a more general identity, which gives
an elegant expansion of the product of a skew Schur function with a quantum
power sum function in terms of skew Schur functions. We give two proofs, one
completely bijective in the spirit of Assaf-McNamara's original proof, and one
via Lam-Lauve-Sotille's skew Littlewood-Richardson rule. We end with some
conjectures for skew rules for Hall-Littlewood polynomials.
