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

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Let f ( z ) = ∑ k = 0 ∞ a k z k a 0 > 0 (0.1) be a formal power series. In 1913, G. Pólya [7] proved that if, for all sufficiently large n, the sections f n ( z ) = ∑ k = 0 n a k z k (0.2) have real negative zeros only, then the series (0.1) converges in the whole complex plane C, and its sum f(z) is an entire function of order 0. Since then, formal power series with restrictions on zeros of their sections have been deeply investigated by several mathematicians. We cannot present an exhaustive bibliography here, and restrict ourselves to the references [1, 2, 3], where the reader can find detailed information. In this paper, we propose a different kind of generalisation of Pólya's theorem. It is based on the concept of multiple positivity introduced by M. Fekete in 1912, and it has been treated in detail by S. Karlin [4].