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Infinite log-concavity for polynomial Pólya frequency sequences


Authors: Petter Brändén and Matthew Chasse
Journal: Proc. Amer. Math. Soc. 143 (2015), 5147-5158
MSC (2010): Primary 05A20, 26C10, 05E99, 30C15
DOI: https://doi.org/10.1090/proc/12654
Published electronically: May 22, 2015
MathSciNet review: 3411133
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Abstract: McNamara and Sagan conjectured that if $ a_0,a_1, a_2, \ldots $ is a Pólya frequency (PF) sequence, then so is $ a_0^2, a_1^2 -a_0a_2, a_2^2-a_1a_3, \ldots $. We prove this conjecture for a natural class of PF-sequences which are interpolated by polynomials. In particular, this proves that the columns of Pascal's triangle are infinitely log-concave, as conjectured by McNamara and Sagan. We also give counterexamples to the first mentioned conjecture.

Our methods provide families of nonlinear operators that preserve the property of having only real and nonpositive zeros.


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Additional Information

Petter Brändén
Affiliation: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Email: pbranden@kth.se

Matthew Chasse
Affiliation: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Email: chasse@kth.se

DOI: https://doi.org/10.1090/proc/12654
Keywords: Log-concavity, infinite log-concavity, real zeros, P\'olya frequency sequence
Received by editor(s): June 23, 2014
Received by editor(s) in revised form: October 14, 2014
Published electronically: May 22, 2015
Additional Notes: The first author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. The research is also supported by the Göran Gustafsson Foundation.
Communicated by: Patricia Hersh
Article copyright: © Copyright 2015 American Mathematical Society