Some Properties on Complex Functional Difference Equations

Abstract

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form ∑λ∈Iαλ(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.