In this paper, we consider possible orders of transcendental meromorphic solutions of linear difference equations \((+) \quad \quad \quad {P}_{m}(z)\Delta^{m}f(z)+\cdots+{P}_{1}(z)\Delta f(z)+{P}_{0}(z)f(z)=0,\) where Pj(z) are polynomials for j = 0, …, m. Firstly, we give the condition on existence of transcendental entire solutions of order less than 1 of difference equations (+). Secondly, we give a list of all possible orders which are less than 1 of transcendental entire solutions of difference equations (+). Moreover, the maximum number of distinct orders which are less than 1 of transcendental entire solutions of difference equations (+) are shown. Further, in both cases, for a given difference equation (+) with polynomial coefficients, we can construct a meromorphic solution of (+) of order ρ(f) = ρ for any ρ ∈ [1, +∞). Thirdly, for any given rational number 0 < ρ < 1, we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order ρ. Lastly, some examples are illustrated for our main theorems.
