On Power Series Having Sections with Multiply Positive Coefficients and a Theorem of Polya

Let (0.1) be a formal power series. In 1913, G. Pólya 7] provedthat if, for all sufficiently large n, the sections (0.2) have real negative zeros only, then the series (0.1) convergesin the whole complex plane C, and its sum f(z) is an entirefunction of order 0. Since then, formal power series with restrictionson zeros of their sections have been deeply investigated byseveral mathematicians. We cannot present an exhaustive bibliographyhere, and restrict ourselves to the references 1, 2, 3], wherethe reader can find detailed information. In this paper, we propose a different kind of generalisationof Pólya's theorem. It is based on the concept of multiplepositivity introduced by M. Fekete in 1912, and it has beentreated in detail by S. Karlin 4].