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- W2247104735 abstract "The bivariate series $theta (q,x):=sum _{j=0}^{infty}q^{j(j+1)/2}x^j$ %(where $(q,x)in {bf C}^2$, $|q|<1$) defines a {em partial theta function}. For fixed $q$ ($|q|<1$), $theta (q,.)$ is an entire function. For $qin (-1,0)$ the function $theta (q,.)$ 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 $theta (bar{q}_k,.)$ has a double real zero $bar{y}_k$ (the rest of its real zeros being simple). For $k$ odd (resp. for $k$ even) $theta (bar{q}_k,.)$ 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. $theta (bar{q}_k,.)$ has a local maximum at $bar{y}_k$ and for $k$ sufficiently large $bar{y}_k$ is the second from the left of the real negative zeros of $theta (bar{q}_k,.)$). For $k$ sufficiently large one has $-1<bar{q}_{k+1}<bar{q}_k<0$. One has $bar{q}_k=1-(pi /8k)+o(1/k)$ and $|bar{y}_k|rightarrow e^{pi /2}=4.810477382ldots$." @default.
- W2247104735 created "2016-06-24" @default.
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- W2247104735 date "2015-04-22" @default.
- W2247104735 modified "2023-09-27" @default.
- W2247104735 title "On a partial theta function and its spectrum" @default.
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