Exponential polynomials as solutions of certain nonlinear difference equations

Recently, C.-.C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn + L(z, f) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)2 +q(z)f(z +1) = p(z), where p(z),q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ ℂ, equations of the form f(z)n +q(z)eQ(z)f(z +c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.