Volume 5, pp. 7-17, 1997.

On a converse of Laguerre's Theorem

Thomas Craven and George Csordas

Abstract

The problem of characterizing all real sequences $\{{\gamma }_k{\}}_{k=0}^{\mathrm{\infty }}$ with the property that if $p(x)=\sum _{k=0}^n{a}_k{x}^k$ is any real polynomial, then $\sum _{k=0}^n{\gamma }_k{a}_k{x}^k$ has no more nonreal zeros than $p(x)$, remains open. Recently, the authors solved this problem under the additional assumption that the sequences $\{{\gamma }_k{\}}_{k=0}^{\mathrm{\infty }}$, with the aforementioned property, can be interpolated by polynomials. The purpose of this paper is to extend this result to certain transcendental entire functions. In particular, the main result establishes a converse of a classical theorem of Laguerre for these transcendental entire functions.

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Key words

Laguerre–Pólya class, entire functions, zero distribution, multiplier sequences.

AMS subject classifications

26C10, 30D15, 30D10.