Expansion in Newton interpolation series and \(U\)-Laplace transform (Q1128249)

It is considered the possibility of the representation of an entire function by a Newton interpolation series. Let \(U= \{u_n, u_0=1\}\), be a sequence of positive, non-zero real numbers, satisfying \(u_{n+1} \geq u_n\) and \(\lim u_n= +\infty\) and \(V= \{v_n\}\) be a sequence of complex numbers such that \(\max_{1\leq i\leq n} | v_i |\leq u_n\). An entire function \(f\) is said to have finite \(U\)-exponential type if there exist two numbers \(A>0\) and \(\alpha>0\) such that for sufficiently large \(r\) \[ \max_{| z|=r} \bigl| f(z)\bigr |\leq A \sum_{n=0 }^\infty {(\alpha r)^n \over \Pi_{1\leq i\leq n} u_i}. \] The main result of this paper is the following Theorem 4.1. If the entire function \(f\) has finite \(U\)-exponential type in the some disk of the radius depending on \(U\) and \(V\) then there exist complex numbers \(A_0,A_1, \dots\) such that for every \(z\in\mathbb{C}\), \(f(z)= \sum^\infty_{n=0} A_nP_n(z)\) where \(P_0=1\), \(P_n(z)= \Pi_{1\leq i\leq n} (z-v_i)\).