The leading root of the partial theta function

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.

LaTeX2e, 22 pages including one Postscript figure. Version 2 includes a few new brief remarks; published in Advances in Mathematics

References in corpus (1)

• A ridiculously simple and explicit implicit function theorem

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