An iterative-bijective approach to asymmetric generalizations of Schur's theorem

In this paper, we present a new Rogers--Ramanujan type identity for overpartitions by extending the asymmetrical version of Schur's theorem due to Lovejoy to a broader class of infinite products. More precisely, we provide a combinatorial interpretation of the following product, for any positive integer k, as a generating function for a class of overpartitions in which parts appear in 2ᵏ - 1 colors: \[ (-y_1 q;q)_∞ ⋯ (-y_k q;q)_∞/(y_1 d q;q)_∞. \] Our proof is bijective and unifies two earlier approaches: Lovejoy's bijective proof of the asymmetrical Schur theorem and the iterative-bijective technique developed by Corteel and Lovejoy.