Iterated sequences and the geometry of zeros

作者：Branden Petter^*

来源：Journal fur die Reine und Angewandte Mathematik, 2011, 658: 115-131.

DOI：10.1515/CRELLE.2011.063

We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of Polya-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 + ... + 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-Polya class, and discuss the consequences to problems on iterated Turan inequalities, studied by Craven and Csordas. Finally, we propose a new approach to a conjecture of Boros and Moll.

• 出版日期2011-9