On conditions for convergence of Taylor-Dirichlet series in a convex domain

Abstract

We establish necessary and sufficient conditions for the absolute convergence of the series $$\mathop \sum \limits_{v = 1}^\infty \sum\limits_{k = 0}^{m_v - I} {a_{v,k} z^k \exp (\lambda _v z)}$$ in an open region. We also give conditions under which an arbitrary function analytic in a closed region (analytic in an open region and continuous in a closed region) can be represented by a series of this type.