Computational Methods and Function Theory 3 (2003), No. 2, 425--441
Copyright Heldermann Verlag 2003
Olga M. Katkova
olga.m.katkova@ilt.kharkov.ua , Department of Mathematics, Kharkov State University, Svobody sq., 4, 310077, Kharkov, Ukraine.
Tetyana Lobova
lobova@web.de , Fakultät für Mathematik und Physik, Eberhard Karls Universität Tübingen, D8Q02, Auf der Morgenstelle 14, 72076 Tübingen.
Anna M. Vishnyakova
anna.m.vishnyakova@univer.kharkov.ua , Department of Mathematics, Kharkov State University, Svobody sq., 4, 310077, Kharkov, Ukraine.
In this paper we investigate the class $A^\ast$ 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 $A^\ast$. The main result of the paper is the complete answer to the question: for which $a$ does the function $g_a(z):=\sum_{k=0}^\infty a^{-k^2}z^k$, $a>1$, have only real zeros.
Keywords: Zeros of entire functions, polynomials with real zeros, totally positive sequences, sections of power series.
MSC 2000: 30C15, 30D15.
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