Uniqueness of entire functions concerning differential-difference polynomials sharing small functions

TL;DR

This paper studies the uniqueness of certain difference polynomials involving entire functions that share small functions, introducing new sharing notions and extending previous results in the field.

Contribution

It introduces weakly and relaxed weighted sharing concepts for difference polynomials and extends existing uniqueness theorems for entire functions.

Findings

Established new uniqueness results for difference polynomials sharing small functions.

Improved and extended previous theorems by Sahoo and Biswas.

Provided conditions under which difference polynomials are identical.

Abstract

In this paper, we investigate the uniqueness problem of difference polynomials $f^{n}(z)P(f(z))L_c(f)$ and $g^{n}(z)P(g(z))L_c(g)$, where $L_c(f)=f(z+c)+c_0f(z)$, $P(z)$ is a polynomial with constant coefficients of degree $m$ sharing a small function with the notions of weakly weighted sharing and relaxed weighted sharing and obtained the corresponding results, which improve and extend some recent results due to Sahoo and Biswas (Tamkang J. Math., \textbf{49}(2), 85--97 (2018)).