Uniqueness of certain type of differential-difference and difference polyniminals

Abstract

In this paper we consider certain difference and differential-difference polynomials sharing some polynomial and improve a number of results in \cite{8a}, \cite{8aa} and \cite{18}. In particular we point out a gap in the argument in the proof of the main results in \cite {8aa} and rectifying the same we improve and extend the result to a large extent.

Uniqueness Results on Q-Difference of Meromorphic Functions Using Weakly Weighted Sharing

“…Pointing out some gaps in the proofs of Theorem D & E, Banerjee and Majumder [2] obtained the correct forms of Theorem D & E. Further, by using the concept of weighted sharing, they have reduced the condition n in both the theorems. Also they have combined the two theorems D & E and presented them as a single theorem stated below.…”

“…Theorem F. [2] Let f (z) and g(z) be transcendental entire functions of finite order and n, k be two positive integers. Suppose that c is a non-zero complex constant such that…”

Results on Product of Difference Operator Related to Entire Functions

“…Definition 6. [1] Let f and g share a IM. Also let k be a positive integer or infinity and a ∈ C ∪ {∞}.…”

“…Lemma 8. [1] Let F and G be non-constant meromorphic functions that share (1, 2) * and H ̸ ≡ 0. Then…”

Uniqueness of entire functions whose difference polynomials share a polynomial with finite weight

“…Using these theories, some mathematicians in the world began to study the uniqueness questions of meromorphic functions sharing values with their shifts, and study the value distribution of the nonlinear difference polynomials, and produced many fine works, for example, see [1,5,6,7,11,15,16,23,27,29,30,31,40,44]. We recall the following result from Laine-Yang [23].…”