Exponential polynomials as solutions of certain nonlinear difference equations (Q1757956)

The aim of this paper is to classify the finite order entire (meromorphic) solutions \(f\) of \[ f(z)^n + q(z)e^{Q(z)}f(z + c) = P(z), \] where \(q,Q,P\) are polynomials, \(n\geq2\) is an integer and \(c\in \mathbb{C}\setminus \{0\}\). The authors show that every entire finite order solution \(f\) of the equation satisfies \(\sigma(f) = \text{deg}(Q)\). Hence such an \(f\) could be an exponential polynomial of the form \(f(z) = P_1(z)e^{Q_1}(z) +\dots+P_k(z)e^{Q_k}(z)\), where \(P_j\) and \(Q_j\) are polynomials in \(z\). It turns out that every solution of this form reduces to a function that belongs to one of two classes of transcendental entire functions. A classification of these solutions in terms of growth and zero distribution is given. The reasoning relies on an earlier paper due to \textit{N. Steinmetz} [Manuscr. Math. 26, 155--167 (1978; Zbl 0395.30023)].