On a partial theta function and its spectrum

The bivariate series $\theta (q,x):={\sum }_{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{x}^j$ %(where $(q,x)\in {\mathbf{C}}^2$, $\mathrm{\mid }q\mathrm{\mid }<1$) defines a {\em partial theta function}. For fixed $q$ ($\mathrm{\mid }q\mathrm{\mid }<1$), $\theta (q,\mathrm{.})$ is an entire function. For $q\in (-1,0)$thefunction$\theta (q,.)$ has infinitely many negative and infinitely many positive real zeros. There exists a sequence $\{{\bar{q}}_j\}$ of values of $q$ tending to $-1^{+}$ such that $\theta ({\bar{q}}_k,\mathrm{.})$ has a double real zero ${\bar{y}}_k$ (the rest of its real zeros being simple). For $k$ odd (resp. for $k$ even) $\theta ({\bar{q}}_k,\mathrm{.})$ has a local minimum at ${\bar{y}}_k$ and ${\bar{y}}_k$ is the rightmost of the real negative zeros of $\theta (\bar{q}_k,.)$(resp\mathrm{.}$\theta (\bar{q}_k,.)$hasalocalmaximumat$\bar{y}_k$andfor$k$sufficientlylarge$\bar{y}_k$ is the second from the left of the real negative zeros of $\theta ({\bar{q}}_k,\mathrm{.})$). For $k$ sufficiently large one has $-1<{\bar{q}}_{k+1}<{\bar{q}}_k<0$. One has ${\bar{q}}_k=1-(\pi \mathrm{/}8k)+o(1\mathrm{/}k)$ and $\mathrm{\mid }{\bar{y}}_k\mathrm{\mid }\to e^{\pi \mathrm{/}2}=4.810477382\dots$.