Some results on uniqueness of an entire function with its shift difference

TL;DR

This paper investigates the uniqueness of entire functions satisfying certain differential difference equations, establishing bounds on their hyper order and proving a conjecture in special cases.

Contribution

It introduces new results linking the hyper order of solutions to differential difference equations and proves a related conjecture in specific cases.

Findings

Solutions have hyper order equal to the degree of P.

Proved the differential difference counterpart of Brück conjecture.

Confirmed the conjecture for hyper order less than 1/2.

Abstract

In the paper, we have shown that every entire solution of the differential difference equation $\Delta_{\eta}^{m}f-Q_1=(\Delta_{\eta}^{k}f-Q_{2})e^{P}$ satisfy hyper order of $f=$degree of $P$ and using this result we prove differential difference counter part Br\"uck conjecture. Also we have proved differential difference counter part Br\"uck conjecture for the special case of hyper order of $f<\frac{1}{2}$.