Weighted sharing of meromorphic functions concerning certain type of linear difference polynomials

In this research article, with the help of Nevanlinna theory we study the uniqueness problems of transcendental meromorphic functions having finite order in the complex plane $\mathbb{C}$, of the form is given by $\phi^{n}(z)\sum_{j=1}^{d}a_{j}\phi(z+c_{j})$ and $\psi^{n}(z)\sum_{j=1}^{d}a_{j}\psi(z+c_{j})$ where $L(z,\phi)=\sum_{j=1}^{d}a_{j}\phi(z+c_{j})$ which share a non-zero polynomial $p(z)$ with finite weight. By considering the concept of weighted sharing introduced by I. Lahiri (Complex Variables and Elliptic equations,2001,241-253), we investigate difference polynomials for the cases $(0,2),(0,1),(0,0)$. Our new findings extends and generalizes some classical results of Sujoy Majumder\cite{m11}. Some examples have been exhibited which are relevant to the content of the paper.