This paper presents a novel extension of the $\{1,2,3,1^{k}\}$-inverse concept to complex rectangular matrices, denoted as a $W$-weighted $\{1,2,3,1^{k}\}$-inverse (or $\{1',2',3',{1^{k}}'\}$-inverse), where the weight $W \in \mathbb{C}^{n \times m}$. The study begins by introducing a weighted $\{1,2,3\}$-inverse (or $\{1',2',3'\}$-inverse) along with its representations and characterizations. The paper establishes criteria for the existence of $\{1',2',3'\}$-inverses and extends the criteria to $\{1'\}$-inverses. It is further demonstrated that $A\in \mathbb{C}^{m \times n}$ 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 establishes various representations for the set $A\{ 1',2',3',{1^{k}}'\}$, including canonical representations derived through singular value and core-nilpotent decompositions. This, in turn, yields distinctive canonical representations for the set $A\{ 1,2,3,{1^{k}}\}$. $\{ 1',2',3',{1^{k}}'\}$-inverse is shown to be unique if and only if it has index $0$ or $1$, reducing it to the weighted core inverse. Moreover, the paper investigates properties and characterizations of $\{1',2',3',{1^{k}}'\}$-inverses, which then results in new insights into the characterizations of the set $A\{ 1,2,3,{1^{k}}\}$.

翻译：本文提出了$\{1,2,3,1^{k}\}$-逆概念到复矩形矩阵的新推广，记为$W$-权重$\{1,2,3,1^{k}\}$-逆（或$\{1',2',3',{1^{k}}'\}$-逆），其中权重$W \in \mathbb{C}^{n \times m}$。研究首先引入加权$\{1,2,3\}$-逆（或$\{1',2',3'\}$-逆）及其表示与刻画。论文建立了$\{1',2',3'\}$-逆的存在性准则，并将该准则推广至$\{1'\}$-逆。进一步证明，$A\in \mathbb{C}^{m \times n}$存在$\{1',2',3',{1^{k}}'\}$-逆当且仅当$r(WAW)=r(A)$，其中$r(\cdot)$表示矩阵的秩。本文还建立了集合$A\{ 1',2',3',{1^{k}}'\}$的多种表示，包括通过奇异值分解和核幂零分解导出的典范表示。进而得到集合$A\{ 1,2,3,{1^{k}}\}$的独特典范表示。$\{ 1',2',3',{1^{k}}'\}$-逆在指数为$0$或$1$时唯一，此时退化为加权核心逆。此外，本文研究了$\{1',2',3',{1^{k}}'\}$-逆的性质与刻画，从而为集合$A\{ 1,2,3,{1^{k}}\}$的刻画提供新见解。