Stabilization of the asymptotic expansions of the 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 $\theta$. These series are of the form $-q^{-j}+(-1)^jq^{j(j-1)/2}(1+\sum _{k=1}^{\infty}g{j,k}q^k)$. The coefficients of the stabilized series are expressed by the positive integers $r_k$ giving the number of partitions into parts of three different kinds. They satisfy the recurrence relation $r_k=\sum {\nu =1}^{\infty}(-1)^{\nu -1}(2\nu +1)r{k-\nu (\nu +1)/2}$. Set $(H_{m,j})~:~(\sum {k=0}^{\infty}r_kq^k) (1-q^{j+1}+q^{2j+3}-\cdots +(-1)^{m-1}q^{(m-1)j+m(m-1)/2})= \sum _{k=0}^{\infty}\tilde{r}{k;m,j}q^k$. Then for $k\leq (m+2j)(m+1)/2-1-j$ and $j\geq (2m-1+\sqrt{8m^2+1})/2$ one has $g_{j,k}=\tilde{r}_{k;m,j}$.