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In the paper, we introduce a new notion of reduced linear c-shift operator $L _{c}^{r}\,f$, and with the aid of this new operator, we study the uniqueness of meromorphic functions $f(z)$ and $L_{c}^{r}\,f$ sharing two or more values in the extended complex plane. The results obtained in the paper significantly improve a number of existing results. Further, using the notion of weighted sharing of sets, we deal the same problem. We exhibit a handful number of examples to justify certain statements relevant to the content of the paper. We are also able to determine the form of the function that coincides with its reduced linear c-shift operator. At the end of the paper, we pose an open question for future research.

Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets