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This paper presents a novel extension of the $\{1,2,3,1^{k}\}$-inverseconcept to complex rectangular matrices, denoted as a $W$-weighted$\{1,2,3,1^{k}\}$-inverse (or $\{1',2',3',{1^{k}}'\}$-inverse), where theweight $W \in \mathbb{C}^{n \times m}$. The study begins by introducing aweighted $\{1,2,3\}$-inverse (or $\{1',2',3'\}$-inverse) along with itsrepresentations and characterizations. The paper establishes criteria for theexistence of $\{1',2',3'\}$-inverses and extends the criteria to$\{1'\}$-inverses. It is further demonstrated that $A\in \mathbb{C}^{m \timesn}$ admits a $\{1',2',3',{1^{k}}'\}$-inverse if and only if $r(WAW)=r(A)$,where $r(\cdot)$ is the rank of a matrix. The work additionally establishesvarious representations for the set $A\{ 1',2',3',{1^{k}}'\}$, includingcanonical representations derived through singular value and core-nilpotentdecompositions. This, in turn, yields distinctive canonical representations forthe set $A\{ 1,2,3,{1^{k}}\}$. $\{ 1',2',3',{1^{k}}'\}$-inverse is shown to beunique if and only if it has index $0$ or $1$, reducing it to the weighted coreinverse. Moreover, the paper investigates properties and characterizations of$\{1',2',3',{1^{k}}'\}$-inverses, which then results in new insights into thecharacterizations of the set $A\{ 1,2,3,{1^{k}}\}$.

A $W$-weighted generalization of $\{1,2,3,1^{k}\}$-inverse for rectangular matrices