On Malmquist type theorem of complex difference equations (Q2862657)

The paper is in the line of development of a theorem due to \textit{J. Malmquist} [Acta Math. 36, 297--343 (1913; JFM 44.0384.01)] on differential equations in the complex domain by extension of the results to their finite difference analogues. The main result is as follows: Consider the difference equation in the complex domain NEWLINE\[NEWLINE\sum_{\lambda\in I}\alpha_{\lambda}(z)\left(\prod_{\nu=1}^nf(\cdot+c_{\nu})^{l_{\lambda,\nu}} \right) = {{P(z,f)}\over{Q(z,f)}}, NEWLINE\]NEWLINE where \(I=\{\lambda=(l_{\lambda,1},\dots,l_{\lambda,n})\mid l_{\lambda,\nu}\in N\cup\{0\},~ \nu=1,\dots,n\}\) is a finite index set, \(P\) and \(Q\) are coprime polynomials in the second argument over the field of the rational functions and \(q:=deg_f Q>0\). Then, if a solution has finitely many poles, it must have the form NEWLINE\[NEWLINE \displaystyle{f(z) = r(z)e^{g(z)} + s(z)} NEWLINE\]NEWLINE with \(r(z)\) and \(s(z)\) being rational functions and \(g(z)\), a transcendental entire function for which there exists a constant \(d\) and the integers \(k_0, k_1,\dots,k_n\) such that NEWLINE\[NEWLINE \displaystyle{k_0g(z) + \sum_{i=1}^n k_ig(z+c_i) = d}. NEWLINE\]