The leading root of the partial theta function
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(n1)/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.
 Publication:

arXiv eprints
 Pub Date:
 June 2011
 arXiv:
 arXiv:1106.1003
 Bibcode:
 2011arXiv1106.1003S
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Complex Variables;
 Mathematics  Number Theory;
 05A15 (Primary);
 05A19;
 05A20;
 05A30;
 05C30;
 11B65;
 11P84;
 30D20;
 33D15;
 33D65 (Secondary)
 EPrint:
 LaTeX2e, 22 pages including one Postscript figure. Version 2 includes a few new brief remarks