The leading root of the partial theta function

TL;DR

This paper investigates the leading root of the partial theta function, proving positivity and negativity properties of its coefficients, revealing new structural insights into its formal power series expansion.

Contribution

It establishes the sign patterns of the coefficients of the leading root of the partial theta function, a novel result in the analysis of this special function.

Findings

All coefficients of -x_0(y) are strictly positive.

Coefficients of -1/x_0(y) after the constant term are strictly negative.

Coefficients of 1/x_0(y)^2 after the constant term are strictly negative except for y^3.

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.