Iterated sequences and the geometry of zeros
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Petter Brändén
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 a0 + a1z + ⋯ + anzn has only real and non-positive zeros, then so does the polynomial 
. 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.
© Walter de Gruyter Berlin · New York 2011
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Artikel in diesem Heft
- K3 surfaces, entropy and glue
- Bäcklund transformations for transparent connections
- Root numbers and parity of ranks of elliptic curves
- Nombres réels de complexité sous-linéaire : mesures d'irrationalité et de transcendance
- On existence of log minimal models II
- Iterated sequences and the geometry of zeros
- Analytic R-groups of affine Hecke algebras
- Strongly free sequences and pro-p-groups of cohomological dimension 2
- Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids
