A property of a partial theta function

The series $\theta (q,x):={\sum }_{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{x}^j$ converges for $\mathrm{\mid }q\mathrm{\mid }<1$ 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 ${\tilde{q}}_1$, ${\tilde{q}}_2$, $\dots$ (where $0.3092493386\dots ={\tilde{q}}_1<{\tilde{q}}_2<\cdots <1$, $\lim _{j\rightarrow \infty}\tilde{q}_j=1$)thefunction$\theta (q,.)$ 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 $\mathcal{L}\mathcal{P}-\mathcal{I}$;
2. for $q\in \mathbb{C}\mathrm{\backslash }0$, $\mathrm{\mid }q\mathrm{\mid }<1$, $\theta (q,x)=\prod _i(1+x/x_i)$,where$-x_i$arethezerosof$\theta$;
3. for any fixed $q\in \mathbb{C}\mathrm{\backslash }0$, $\mathrm{\mid }q\mathrm{\mid }<1$, 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 $\mathcal{L}\mathcal{P}$.
5. for any fixed $q\in \mathbb{C}\mathrm{\backslash }0$, $\mathrm{\mid }q\mathrm{\mid }<1$, 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 $\theta$.