This paper presents a novel extension of the {1,2,3,1k}\documentclass[12pt]{minimal}
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\begin{document}$$\{1,2,3,1^{k}\}$$\end{document}-inverse concept to complex rectangular matrices, denoted as a W-weighted {1,2,3,1k}\documentclass[12pt]{minimal}
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\begin{document}$$\{1,2,3,1^{k}\}$$\end{document}-inverse, using a complex rectangular matrix W. The study begins by introducing a weighted {1,2,3}\documentclass[12pt]{minimal}
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\begin{document}$$\{1,2,3\}$$\end{document}-inverse along with its representations and characterizations. The paper establishes criteria for the existence of the proposed inverses based on rank equalities and extends it to weighted inner inverses. The work additionally establishes various representations, properties and characterizations of W-weighted {1,2,3,1k}\documentclass[12pt]{minimal}
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\begin{document}$$\{1,2,3,1^{k}\}$$\end{document}-inverses, including canonical representations derived through singular value and core-nilpotent decompositions. This, in turn, yields distinctive canonical representations and characterizations of the set A{1,2,3,1k}\documentclass[12pt]{minimal}
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\begin{document}$$A\{ 1,2,3,{1^{k}}\}$$\end{document}. Furthermore, it is shown that W-weighted {1,2,3,1k}\documentclass[12pt]{minimal}
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\begin{document}$$\{1,2,3,1^{k}\}$$\end{document}-inverse is unique if and only if it has index 0 or 1, reducing it to the weighted core inverse.
