Asymptotic expansions of zeros of a partial theta function

TL;DR

This paper investigates the zeros of the partial theta function, proving their Laurent series coefficients stabilize and are positive integers, satisfying a specific recurrence relation, thus revealing new structural properties of these zeros.

Contribution

It establishes the stabilization and positivity of Laurent series coefficients of the zeros of the partial theta function, along with their recurrence relation.

Findings

Coefficients of zeros' Laurent series stabilize and are positive integers.

The stabilized coefficients follow a specific recurrence relation.

The results reveal new structural insights into the zeros of the 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 $r_k=\sum _{\nu =1}^{\infty}(-1)^{\nu -1}(2\nu +1)r_{k-\nu (\nu +1)/2}$.