Convergence domain for series of exponential polynomials

Abstract

In this paper we study the convergence of exponential polynomials series constructed by almost exponential sequences of such polynomials. Particular cases of such series are series of exponential monoms, exponential series, Dirichlet series and power series. We obtain an analogue of Abel theorem for these series implying in particular results on continuation of convergence. An analogue of the Cauchy–Hadamard theorem is obtained as well. We give a formula allowing one to recover the convergence domain for these series by their coefficients. The obtained results include Abel and Cauchy–Hadamard theorems for exponential monoms series, exponential series, Dirichlet series and power series.