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 $r_k={\sum }_{\nu =1}^{\infty }(-1{)}^{\nu -1}(2\nu +1)r_{k-\nu (\nu +1)/2}$.

References in corpus (3)

• Two truncated identities of Gauss
• Hardy-Petrovitch-Hutchinson's problem and partial theta function
• The leading root of the partial theta function