Asymptotic expansions of zeros of a partial theta function

Abstract

The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the Laurent series in $q$ of the zeros of $\theta$. The coefficients $r_k$ of the stabilized series are positive integers. They are the elements of a known increasing sequence satisfying the recurrence relation $r_k=\sum _{\nu =1}^{\infty}(-1)^{\nu -1}(2\nu +1)r_{k-\nu (\nu +1)/2}$.