Parametrizing $W$-weighted BT inverse to obtain the $W$-weighted $q$-BT inverse

The core-EP and BT inverses for rectangular matrices were studied recently in the literature. The main aim of this paper is to unify both concepts by means of a new kind of generalized inverse called $W$-weighted $q$-BT inverse. We analyze its existence and uniqueness by considering an adequate matrix system. Basic properties and some interesting characterizations are proved for this new weighted generalized inverse. Also, we give a canonical form of the $W$-weighted $q$-BT inverse by means of the weighted core-EP decomposition.