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Positivity of Szegö's rational function

Positivity of Szegö's rational function
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.

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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},
}