The leading root of the partial theta function
Combinatorics
2012-02-07 v2 Mathematical Physics
Classical Analysis and ODEs
Complex Variables
math.MP
Number Theory
Abstract
I study the leading root x_0(y) of the partial theta function \Theta_0(x,y) = \sum_{n=0}^\infty x^n y^{n(n-1)/2}, considered as a formal power series. I prove that all the coefficients of -x_0(y) are strictly positive. Indeed, I prove the stronger results that all the coefficients of -1/x_0(y) after the constant term 1 are strictly negative, and all the coefficients of 1/x_0(y)^2 after the constant term 1 are strictly negative except for the vanishing coefficient of y^3.
Cite
@article{arxiv.1106.1003,
title = {The leading root of the partial theta function},
author = {Alan D. Sokal},
journal= {arXiv preprint arXiv:1106.1003},
year = {2012}
}
Comments
LaTeX2e, 22 pages including one Postscript figure. Version 2 includes a few new brief remarks; published in Advances in Mathematics