Uniqueness of differential-difference polynomials of meromorphic functions sharing shift polynomial and \\small function IM

Abstract

In this present research article, we investigate some uniqueness problems of meromorphic functions concerning their product of differential polynomials sharing $P(h(z))\displaystyle\prod_{j=1}^{s}h(z+c_{j})^{\mu_{j}}$ as the generalized shift polynomial and small function $IM$. We consider $f(z)$ and $g(z)$ be two non constant finite-order meromorphic functions, if $\left[f_{1}^{p} \mathcal{P}\left(f_{1}\right)\Phi(z) \right]^{(k)}$ and $\left[g_{1}^{p} \mathcal{P}\left(g_{1}\right) \Psi(z)\right]^{(k)}$ share $\alpha(z)$ IM, $f(z)$ and $g(z)$ share $\infty$ IM, then $f_{1} \equiv \operatorname{tg}_{1}$ or $f_{1} \equiv \operatorname{g}_{1}$ for a constant $t$. The results of the paper improves and generalizes the several earlier results of Molla Basir Ahamed and Goutam Haldar.