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 $-q^{-j}+(-1{)}^j{q}^{j(j-1)/2}(1+{\sum }_{k=1}^{\infty }{g}_{j,k}{q}^k)$. 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 $(H_{m,j}):({\sum }_{k=0}^{\infty }{r}_k{q}^k)(1-q^{j+1}+q^{2j+3}-\cdots +(-1{)}^{m-1}{q}^{(m-1)j+m(m-1)/2})={\sum }_{k=0}^{\infty }{\tilde{r}}_{k;m,j}{q}^k$. Then for $k\le (m+2j)(m+1)/2-1-j$ and $j\ge (2m-1+\sqrt{8m^2+1})/2$ one has $g_{j,k}={\tilde{r}}_{k;m,j}$.

Cited by in corpus (1)

• Uniform bounds on locations of zeros of partial theta function