Armin Straub — Advances in Applied Mathematics — Volume 41, Number 2, 2008, Pages 255-264
Abstract
We consider the problem of deciding whether a given rational function has a power series expansion with all its coefficients positive. Introducing an elementary transformation that preserves such positivity we are able to provide an elementary proof for the positivity of Szegö's function \[ \frac{1}{(1 - x) (1 - y) + (1 - y) (1 - z) + (1 - z) (1 - x)} \] which has been at the historical root of this subject starting with Szegö. We then demonstrate how to apply the transformation to prove a \(4\)-dimensional generalization of the above function, and close with discussing the set of parameters \((a, b)\) such that \[ \frac{1}{1 - (x + y + z) + a (xy + yz + zx) + bxyz} \] has positive coefficients.Download
| Link | Size | Description | Hits |
|---|---|---|---|
| positivity.pdf | 140.76 KB | Positivity of Szegö's rational function (PDF, 9 pages) | 3569 |
| positivity-addendum.pdf | 50.22 KB | Addendum to "Positivity of Szegö's rational function" (PDF, 1 page) | 2249 |
BibTeX
@article{positivity-2008,
author = {Armin Straub},
title = {Positivity of {S}zeg\"o's rational function},
journal = {Advances in Applied Mathematics},
year = {2008},
volume = {41},
number = {2},
pages = {255--264},
doi = {10.1016/j.aam.2007.10.001},
}