Partitioning into subclasses of a class of functions that can be represented by subsequences of generalized Dirichlet polynomials (Q795950)

Let \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n,\quad a_ n\neq 0,\quad n=1,2,...\) be an entire function of order \(\rho>0\) and of type \(\sigma>0\) and let \(\{\lambda_ n\}\) be such a sequence of complex numbers that \(\overline{\lim}_{n\to \infty}n/| \lambda_ n|^{\rho}<\infty.\) Let \(H(\{\lambda_ n\})\) denote the class of functions F(z) which can be represented as \[ F(z)=\lim_{n\to \infty}\sum^{n}_{k=1}a_ k^{(n)}f(\lambda_ kz),\quad | z|<\infty, \] where the limit sign means the limit in the sense of uniform convergence on every compact set in \({\mathbb{C}}\). In the reviewed paper it is given the condition on sequences \(\{ \lambda\) '\({}_ n\}\) and \(\{ \lambda\) ''\({}_ n\}\) under which the following equality holds: \(H(\{\lambda '\!_ n\}\cup \{\lambda ''\!_ n\})=H(\{\lambda '\!_ n\})+H(\{\lambda ''\!_ n\}).\)