Effect of the zeros of initial term non-gap polynomials of higher index on the uniqueness of meromorphic functions

In this article, we have first introduced the definition of index of initial term non-gap polynomials (ITNGP). Next, we extend the well known definition of weighted sharing of sets in a more comprehensive manner to establish the definition of weighted sharing of sets in the wider sense. As we have observed that the previous researchers were confined their investigations up to index 3 of ITNGP, we feel that it is reasonable mostly to explore the contribution of ITNGP of index \(\ge 4\) $\ge 4$ in the literature. Thus in this paper we focus to establish the uniqueness of meromorphic functions under the periphery of extended structures of weighted range sets solely from \({\mathbb {C}}\) $\mathbb{C}$ . Consequently, we present four theorems which are totally new of its genre.