To the question on representation of analytic functions by series of generalized exponents (Q1821903)

Let \(f(z)=\sum^{\infty}_{0}\frac{a_ n}{n!}z^ n\) be an entire function of exponential type and completely regular growth. In this paper necessary and sufficient conditions for coefficients \(a_ n\) are obtained for an arbitrary analytic function \(\Phi\) (z) on a convex domain to have a representation of the form \[ (1)\quad \Phi (z)=\sum A_ nf(\lambda_ nz) \] where the complex numbers \(\lambda_ n\) are such that \(\lim_{n\to \infty}\frac{\ell n n}{\lambda_ n}=0.\) Furthermore, necessary and sufficient conditions for \(a_ n\) are obtained for an entire function F(z) with given estimation of the indicator to have a representation (1). Now we formulate one of the results. Theorem 1. An arbitrary analytic function on a convex domain \({\mathcal D}(0\in {\mathcal D}\) has a representation of the form (1) if and only if \(a_ n\neq 0\) for all n and all the singularities of \(\sum (a_ n/t^{n+1}),\sum (1/a_ nt^{n+1})\) are in [0,1]. These resuls strengthen the results of the author from his previous works.