Meromorphic functions sharing the zeros of lower degree symmetric polynomials in weighted wider sense

In this paper, we establish some mathematical rules for determining the initial and terminal numbers of non-zero terms in any arbitrary polynomial. These rules lead to the definitions of index $s$ and reverse index $\hat{s}$ of a polynomial. Further, building on these concepts, we introduce the order of a polynomial $P(z)$ as $(s,\hat{s})$. If $P_{*}(z)$ is another polynomial of order $(\hat{s},s)$, then the pair $P(z)$ and $P_{*}(z)$ are referred to as symmetric polynomials. The concept of symmetric polynomials is central in this work, as we investigate the effects of weighted sharing in the wider sense (see Adv. Stud: Euro-Tbilisi Math. J., 16(4)(2023), 175-189) on the zeros of symmetric polynomials along with the sharing of poles. Our study focuses on symmetric polynomials of degree 3, analyzing their intrinsic properties. Sharing of zeros of polynomials of lower degree are critical in nature and at the same time it exhibits sophisticated structural characteristics, making them an ideal subject for such analysis. Our exploration of the sharing of zeros of symmetric polynomials establishes connections between two non-constant meromorphic functions. The article includes examples of both a general nature and specific, partial cases that serve to illustrate and validate our theoretical results.