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 $2^k - 1$ colors: $$\frac{(-y_1 q;q)_\infty \cdots (-y_k q;q)_\infty}{(y_1 d q;q)_\infty}.$$ 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.