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On Power Series Having Sections With Only Real Zeros

On Power Series Having Sections With Only Real Zeros In this paper we investigate the class A* of power series with non-negative coefficients such that all but a finite number of its sections have only real zeros. We obtain some new necessary conditions for a power series to belong to A*. The main result of the paper is the complete answer to the question: for which a does the function \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$g_a(z):=\sum_{k=0}^\infty a^{-k^2}z^k$\end{document}, a >1 have only real zeros. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On Power Series Having Sections With Only Real Zeros

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Publisher
Springer Journals
Copyright
Copyright © Heldermann  Verlag 2003
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/bf03321047
Publisher site
See Article on Publisher Site

Abstract

In this paper we investigate the class A* of power series with non-negative coefficients such that all but a finite number of its sections have only real zeros. We obtain some new necessary conditions for a power series to belong to A*. The main result of the paper is the complete answer to the question: for which a does the function \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$g_a(z):=\sum_{k=0}^\infty a^{-k^2}z^k$\end{document}, a >1 have only real zeros.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 1, 2004

Keywords: Zeros of entire functions; polynomials with real zeros; totally positive sequences; sections of power series; 30C15; 30D15

References