Iterated sequences and the geometry of zeros
By Petter Brand'en. 2009
63 citations recorded by the source corpus.
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 a 2 0 +(a 1 − a0a2)z+ · · ·+ (a n−1 − an−2an)z + a2nz 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–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.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/42c3336248efbe57bbe7c81b9a285df8dc52b45d
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Higher order Turán inequalities for the partition function (2017, methodology): “It should be noted that the above notion of higher order iterated Turán inequalities coincides with the notion of the higher order log-concavity introduced by Moll [34], see also [3, 4].”
- Hyperbolicity and stable polynomials in combinatorics and probability (2012, background): “In a brilliant series of papers [BB09a, BB09b, BB10], Borcea and Brändén finished off the problem of extending the Pólya-Schur Theorem to multivariate stability preserving maps.”