Entire solutions of a certain type differential-difference equation and differential-difference analogue of Bruck conjecture

In the paper, we find out the precise form of the finite order entire solutions of the following differential-difference equation $$f^{(k)}(z)=\text{sideset}{}_{j=0}^n\sum a_{j}f(z+jc),$$ where $a_0,a_1,\dots ,a_n(\mathrm{\ne }0)\in \mathbb{C}$. Also in the paper we study the differential-difference analogue of Brück conjecture and derive a uniqueness result of finite order entire function $f(z)$ having a Borel exceptional small function of $f(z)$, when $f^{(k)}(z)$ and $\text{sideset}{}_{j=0}^n\sum a_{j}f(z+jc)$ share a small function of $f(z)$. The obtained results, significantly generalize and improve the results due to Liu and Dong (Some results related to complex differential-difference equations of certain types, Bull. Korean Math. Soc., 51 (5) (2014), 1453-1467). Some examples are given to ensure the necessity of the condition (s) of our main results.