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Journal of the London Mathematical Society 1998 58(1):97-110; doi:10.1112/S0024610798006280
© 1998 by London Mathematical Society
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© The London Mathematical Society

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

I. V. Ostrovskii and N. A. Zheltukhina

Department of Mathematics, Bilkent University 06533 Bilkent, Ankara, Turkey. E-mail: iossif{at}fen.bilkent.edu.tr
B. I. Verkin Institute for Low Temperature Physics and Engineering 310164 Kharkov, Ukraine

Received 30 June 1995. Revision received 30 January 1996.

Let


Formula
(0.1)

be a formal power series. In 1913, G. Pólya [7] proved that if, for all sufficiently large n, the sections


Formula
(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].


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