Iterated sequences and the geometry of zeros

Details Iterated sequences and the geometry of zeros Iterated sequences and the geometry of zeros

2009-09-10

arxiv.org/abs/0909.1927v3

J. Reine Angew. Math. 658 (2011), 115-131

Mathematics

Combinatorics

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

Scientific paper

10.1515/CRELLE.2011.063

We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of P\'olya--Schur type are given of the transformations that preserve the property of having only real and non-positive zeros. In particular, if a polynomial $a_0+a_1z +\cdots+a_nz^n$ has only real and non-positive zeros, then so does the polynomial $a_0^2+ (a_1^2-a_0a_2)z+...+ (a_{n-1}^2-a_{n-2}a_n)z^{n-1}+a_n^2z^n$. 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-P\'olya class, and discuss the consequences to problems on iterated Tur\'an inequalities, studied by Craven and Csordas. Finally, we propose a new approach to a conjecture of Boros and Moll.

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