Property of meromorphic solutions of certain high order difference equations

By utilizing the difference analogue of Nevanlinna's value distribution theory of meromorphic functions, the exponents of convergence of zeros, poles and the order of growth of meromorphic solutions of the nonlinear high order difference equation $P_{1}(z)\prod_{i=1}^{n}f(z+c_{i})=P_{2}(z)f(z)^{n}$ are studied, where $n$ is a positive integer, $~c_{i}(i=1,...,n)$ are non-vanishing complex constants, and$~P_{1}(z),~P_{2}(z)$ are given non-vanishing polynomials. The accurate estimate of the order of growth of meromorphic solutions to this difference equation is attained under the given conditions.