A property of a partial theta function

Abstract

The series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for $|q|<1$ and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. It is known that for $q$ taking one of the {\em spectral} values $\tilde{q}_1$, $\tilde{q}_2$, $\ldots$ (where $0.3092493386\ldots =\tilde{q}_1<\tilde{q}_2<\cdots <1$, $\lim _{j\rightarrow \infty}\tilde{q}_j=1$) the function $\theta (q,.)$ has a double zero which is the rightmost of its real zeros (the rest of them being simple). For $q\neq \tilde{q}_j$ the partial theta function has no multiple real zeros. We prove that: 1) for $q\in (\tilde{q}_{j},\tilde{q}_{j+1}]$ the function $\theta$ is a product of a degree $2j$ real polynomial without real roots and a function of the Laguerre-P\'olya class $\cal{LP-I}$; 2) for $q\in \mathbb{C}\backslash 0$, $|q|<1$, $\theta (q,x)=\prod _i(1+x/x_i)$, where $-x_i$ are the zeros of $\theta$; 3) for any fixed $q\in \mathbb{C}\backslash 0$, $|q|<1$, the function $\theta$ has at most finitely-many multiple zeros; 4) for any $q\in (-1,0)$ the function $\theta$ is a product of a real polynomial without real zeros and a function of the Laguerre-P\'olya class $\cal{LP}$. 5) for any fixed $q\in \mathbb{C}\backslash 0$, $|q|<1$, and for $k$ sufficiently large, the function $\theta$ has a zero $\zeta _k$ close to $-q^{-k}$. These are all but finitely-many of the zeros of $\theta$.