Uniqueness of certain polynomials of meromorphic functions sharing a set with finite weight

We investigated the Brück type conjecture and generalized the existing result by extending them up to a difference-differential polynomial \(\mathcal{P}[\xi]\) sharing a small function with a certain differential polynomial \(\pounds[\xi]\) of a meromorphic function. The class of all meromorphic solutions of the differential equation \(\mathcal{P}[\xi] \equiv \pounds[\xi]\) has been explored. Our result will generalize and extend the result due to A. Banerjee and B. Chakraborty [3]. For the generalization of our main result, some relevant questions have finally been posed for further study in this direction.