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ON POWER SERIES HAVING SECTIONS WITH MULTIPLY POSITIVE COEFFICIENTS AND A THEOREM OF PÓLYA

Published online by Cambridge University Press:  01 August 1998

I. V. OSTROVSKII  and
N. A. ZHELTUKHINA
I. V. OSTROVSKII
Affiliation:
Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey. E-mail: [email protected]
N. A. ZHELTUKHINA
Affiliation:
B. I. Verkin Institute for Low Temperature Physics and Engineering, 310164 Kharkov, Ukraine
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Abstract

Let

formula here

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

formula here

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].

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 58 , Issue 1 , August 1998 , pp. 97 - 110
Copyright
The London Mathematical Society 1998

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