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In this paper, we study the difference equation $${a_1}\left(z \right)f\left({z + 1} \right) + {a_0}\left(z \right)f\left(z \right) = 0,$$ where a1(z) and a0(z) are entire functions of finite order. Under some conditions, we obtain some properties, such as fixed points, zeros etc., of the differences and forward differences of meromorphic solutions of the above equation.

Let f (z) be a meromorphic functions with finite order , R(z) be a nonconstant rational function and k be a positive integer. In this paper, we consider the difference equation originated from Schwarzian differential equation, which is of form [?3f(z)?f(z)- 3/2(?2|(z))2]k = R(z)(?f (z))2k. We investigate the uniqueness of meromorphic solution f of difference Schwarzian equation if f shares three values with any meromrphic function. The exact forms of meromorphic solutions f of difference Schwarzian equation are also presented.

In this paper, we mainly study zeros and poles of the forward differences Δnf(z), where f(z) is a finite order meromorphic function with two Borel exceptional values.

Consider the difference equation $$\biggl[\frac{\Delta^{3}f(z)}{\Delta f(z)}-\frac{3}{2} \biggl(\frac {\Delta^{2}f(z)}{\Delta f(z)} \biggr)^{2} \biggr]^{k} =\frac{P(z,f(z))}{Q(z,f(z))}, $$ where $P(z,f)$ and $Q(z,f)$ are prime polynomials in $f(z)$ with $\deg_{f}P=p, \deg_{f}Q=q$ , and $d=\max\{p,q\}>0$ . We give the supremum of d, an estimation of the sum of Nevanlinna exceptional values of meromorphic solution $f(z)$ of the equation, and study the value distributions of their difference $\Delta f(z)$ and divided difference $\frac{\Delta f(z)}{f(z)}$ .