Growth of solutions of some higher order linear difference equations (Q351260)

The authors consider the following difference equation in the complex domain NEWLINE\[NEWLINEf(z+n) = \sum_{j=0}^{n-1}\{P_j(e^z) + Q_j(e^{-z})\}f(z+j) = 0,NEWLINE\]NEWLINE where \(P_j(z)\) and \(Q_j(z)\) (\(j=0,\dots,n-1\)) are polynomials. The following main results are obtained {\parindent=6mm \begin{itemize}\item[1.] If \(deg(P_0)>deg(P_j)\) or \(deg(Q_0)>deg(Q_j)\), \(j=1,\dots,n-1\), then each non-trivial meromorphic solution \(f(z)\) of finite order of the above equation satisfies \(\sigma(f)=\lambda(f-a)\geq 2\), where \(\sigma(f)\) is the order of the meromorphic function \(f\) and \(\lambda(f)\) the exponent of convergence of its zeros; \(a\) is any non-zero complex number. \item[2.] Under the above assumptions, if \(f(z)\) is a non-trivial entire solution of finite order of the above equation that satisfies \(\lambda(f)<1\), then \(\sigma(f)=2\). \item[3.] Under the same assumptions, if \(A(z)\) is a transcendental entire function, every solution of NEWLINE\[NEWLINEf(z+n) = \sum_{j=0}^{n-1}\{P_j(e^{A(z)}) + Q_j(e^{-A(z)})\}f(z+j) = 0NEWLINE\]NEWLINE is of infinite order and \(\sigma_2(f)\geq\sigma(A)\), where \(\sigma_2(f)\) is the hyper-order of the meromorphic function \(f\).NEWLINENEWLINE\end{itemize}}