On properties of meromorphic solutions for Complex Difference Equation of Malmquist type

In this paper, we present the properties on zeros, fixed points, poles, Borel exceptional value of finite order transcendental meromorphic solutions of complex difference equationof Malmquist type ∑j=1nf(z+cj)=R(f(z))=P(f(z))Q(f(z))=apf(z)p+ap−1f(z)p−1+⋯+a1(f(z)+a0bqf(z)q+aq−1f(z)q−1+⋯+b1(f(z)+b0, where n(∈ ℕ) ≥ 2),n(∈ ℕ) ≥ 2),P(f(z)) and Q(f(z)) are relatively prime polynomials in f(z) with rational coefficients as (s = 0, 1, …, p) and bt (t = 0, 1, …, q) such that a0apbq ≢ 0, and also consider the existence and the forms on rational solutions of this type of difference equations. Some examples are also listed to show that the assumptions of theorems, in certain senses, are the best possible.