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On Power Series Having Sections with Multiply Positive Coefficients and a Theorem of Pólya
Author(s) -
Ostrovskii I. V.,
Zheltukhi. A.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006280
Subject(s) - series (stratigraphy) , power series , mathematics , formal power series , order (exchange) , combinatorics , complex plane , function (biology) , plane (geometry) , power (physics) , discrete mathematics , pure mathematics , mathematical analysis , physics , geometry , quantum mechanics , paleontology , finance , evolutionary biology , economics , biology
Let 0.1 f ( z ) = ∑ k = 0 ∞a k z k a 0 > 0be a formal power series. In 1913, G. Pólya [ 7 ] proved that if, for all sufficiently large n , the sections 0.2f n ( z ) = ∑ k = 0 na k z khave 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 ].