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Iterated sequences and the geometry of zeros
Department of Mathematics, Stockholm University.ORCID iD: 0000-0003-1055-1474
2011 (English)In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, ISSN 0075-4102, Vol. 658, 115-131 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2011. Vol. 658, 115-131 p.
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URN: urn:nbn:se:kth:diva-78671DOI: 10.1515/CRELLE.2011.063ISI: 000294610400006OAI: diva2:492753
QC 20120221Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2012-02-21Bibliographically approved

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Brändén, Petter
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