Entire solutions of a differential-difference equation of certain type and the differential-difference analog of the Brück conjecture

Abstract

UDC 517.53; 517.92

We determine the precise form of  finite-order entire solutions of the following differential-difference equation: \begin{align*}f^{(k)}(z)= \sum^n_{j=0} a_j f(z+jc),\end{align*} such that the exponent of convergence of the zeros of $f(z)$ is less than the order of $f(z),$ where $a_0, a_1,\ldots,a_n\,({\neq\,}0)\in\mathbb{C}.$ We also study the differential-difference analog of the Brück conjecture and derive a uniqueness result for a finite-order entire function  $f(z)$  with a Borel exceptional small function of $f(z),$ when $f^{(k)}(z)$ and $\displaystyle\sum\nolimits^n_{j=0} a_j f(z+jc)$ share a small function of $f(z).$ The obtained results significantly generalize and improve the results due to Liu and Dong [Bull. Korean Math. Soc., 51 (5), 1453–1467 (2014)]. Some examples are given to ensure the necessity of the condition (s) of our main results.