On Growth of Meromorphic Solutions of Complex Functional Difference Equations

The main purpose of this paper is to investigate the growth order of the meromorphic solutions of complex functional difference equation of the form (Sigma(lambda is an element of I)alpha(lambda)(z) (Pi(n)(nu=1)integral(z + c(v))(t lambda,nu)))/(Sigma(mu is an element of J)beta(mu) (z)(Pi(n)(nu=1)integral(z + c(nu)(m mu,nu))) = Q(z, integral(p(z))), where I = {lambda = (l(lambda,1),l(lambda,2),...,l(lambda,n)) vertical bar l(lambda,nu) is an element of N boolean OR{0},] nu = 1, 2,...,n} and J = {mu = (m(mu,1),m(mu,2),...,m(mu,n)) vertical bar m(mu,nu) is an element of N boolean OR{0}, nu = 1, 2,...,n} are two finite index sets, c(nu) (nu = 1, 2,...,n) are distinct complex numbers, alpha lambda (z) (lambda is an element of I) and beta(mu) (z) (mu is an element of J) are small functions relative to f(z), and Q(z,u) is a rational function in u with coefficients which are small functions of integral(z), p(z) = p(k)z(k) + p(k-1)z(k-1) + ... + p(0) is an element of C[z] of degree k >= 1. We also give some examples to show that our results are sharp.