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arXiv · 1106.1003

The leading root of the partial theta function

Alan D. Sokal
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Abstract

I study the leading root x_0(y) of the partial theta function Θ_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.

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Alan D. Sokal. 2012-02-06. The leading root of the partial theta function. https://doi.org/10.1016/j.aim.2012.01.012

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