Possible orders of transcendental meromorphic solutions of linear difference equations with polynomial coefficients

In this paper, we consider possible orders of transcendental meromorphic solutions of linear difference equations (+) P-m(z)Delta(m)f(z)+& ctdot;+P1(z)Delta f(z)+P0(z)f(z)=0, where P-j(z) are polynomials for j = 0, & mldr;, 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 rho(f) = rho for any rho is an element of [1, +infinity). Thirdly, for any given rational number 0 < rho < 1, we can construct a linear difference equation with polynomial coefficients which has a transcendental entire solution of order rho. Lastly, some examples are illustrated for our main theorems.