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∈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)), whereIis a finite set of multi-indexesλ=(λ0,λ1,…,λn),c0=0,cj∈ℂ∖{0} (j=1,2,…,n)are distinct complex constants,p(z)is a polynomial, andαλ(z) (λ∈I),ai(z) (i=0,1,…,s), andbj(z) (j=0,1,…,t)are small meromorphic functions relative tof(z). We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.