Some Properties on Complex Functional Difference Equations

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form ∑λ∈I αλ(z)(∏j=0nf(z+cj)λj)=R(z,f∘p)=((a0(z)+a1(z)(f∘p)+ ⋯ +as(z) (f∘p) s)/(b0(z) + b1(z)(f∘p)+ ⋯ +bt(z)(f∘p) t)), where I is a finite set of multi‐indexes λ = (λ0, λ1, …, λn), c0 = 0, cj ∈ ℂ∖{0} (j = 1,2, …, n) are distinct complex constants, p(z) is a polynomial, and αλ(z) (λ ∈ I), ai(z) (i = 0,1, …, s), and bj(z) (j = 0,1, …, t) are small meromorphic functions relative to f(z). We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.