Abstract:

In this paper, we discuss the following difference equations

a_n(z)f(z+n)^j_n+…+a₁(z)f(z+1)^j₁+a₀(z)f(z)^j₀=b(z),

a_n(z)f(qⁿz)^j_n+…+a₁(z)f(qz)^j₁+a₀(z)f(z)^j₀=b(z),

where a_i(z)(i=0, 1, …, n) and b(z) are nonzero rational functions, j_i(i=0, 1, …, n) are positive integers, q is a nonzero complex constant. When the equations above have meromorphic solutions with hyper order less than 1 and few poles, we investigate the distributions of zeros. Besides, when the solution has infinitely many poles, we give the lower bound of the exponent of convergence of poles.

Key words: Nonlinear difference equation, Meromorphic solution, Poles, Zeros, Deficiency