The growth of transcendental entire solutions of linear difference equations with polynomial coefficients

In this paper, we study the growth of transcendental entire solutions of linear difference equations \begin{equation} P_m(z)\Delta^mf(z)+\cdots+P_1(z)\Delta f(z)+P_0(z)f(z)=0,\tag{+} \end{equation} where $P_j(z)$ are polynomials for $j=0,\ldots,m$. At first, we reveal type of binomial series in terms of its coefficients. Second, we give a list of all possible orders, which are less than 1, and types of transcendental entire solutions of linear difference equations $(+)$. In particular, we give so far the best precise growth estimate of transcendental entire solutions of order less than 1 of $(+)$, which improves results in [3, 4], [5], [7]. Third, for any given rational number $\rho\in(0,1)$ and real number $\sigma\in(0,\infty)$, we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order $\rho$ and type $\sigma$. At last, some examples are illustrated for our main theorem.