Iterated sequences and the geometry of zeros

Abstract

We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of Pólya--Schur type are given of the transformations that preserve the property of having only real and non-positive zeros. In particular, if a polynomial $a_0+a_{1}z+\cdots +a_n{z}^n$ has only real and non-positive zeros, then so does the polynomial $a_0^2+(a_1^2-a_0{a}_2)z+...+(a_{n-1}^2-a_{n-2}{a}_n)z^{n-1}+a_n^2{z}^n$. This confirms a conjecture of Fisk, McNamara-Sagan and Stanley, respectively. A consequence is that if a polynomial has only real and non-positive zeros, then its Taylor coefficients form an infinitely log-concave sequence. We extend the results to transcendental entire functions in the Laguerre-Pólya class, and discuss the consequences to problems on iterated Turán inequalities, studied by Craven and Csordas. Finally, we propose a new approach to a conjecture of Boros and Moll.

15 pages. To appear in J. Reine Angew. Math. (Crelle's journal)

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