On A Converse Of Laguerre's Theorem

The problem of characterizing all real sequences ffl k g 1 k=0 with the property that if p(x) = P n k=0 a k x k is any real polynomial, then P n k=0 fl 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 ffl k g 1 k=0 , 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