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On the double zeros of a partial theta function (1504.05786v1)
Published 22 Apr 2015 in math.CA
Abstract: The series $\theta (q,x):=\sum _{j=0}{\infty}q{j(j+1)/2}xj$ converges for $q\in [0,1)$, $x\in \mathbb{R}$, and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. 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 $y_j$ 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 $\tilde{q}_j=1-\pi /2j+(\log j)/8j2+O(1/j2)$ and $y_j=-e{\pi}e{-(\log j)/4j+O(1/j)}$.