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A well-known conjecture (now a theorem) of P. Borwein (see Wang and Krattenthaler - An asymptotic approach to Borwein-type sign pattern theorems) states: For all positive integers $n$, the sign pattern of the coefficients in the expansion of the polynomial $$P_n(q) := (1 −q)(1 −q^2)(1 −q^4)(1 −q^5) ···(1 −q^{3n−2})(1 −q^{3n−1})$$ is ${+}{-}{-}{+}{-}{-}\cdots$, with a coefficient $0$ being considered as both $+$ and $−$.

I'm considering a variant of this problem. Define the polynomial $$Q_n(q):=\prod_{k=1}^n\frac{(1-q^{3k-2})(1-q^{3k-1})}{1-q}.$$ Based on some experiments, I would like to ask:

CLAIM. The coefficients of $Q_n(q)$ make exactly $n$ change of signs, running in blocks of almost equal sizes.

Convention: a coefficient $0$ is treated as both $+$ and $-$.

QUESTION: Is the claim true?

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    $\begingroup$ Do you have an off-by-one error? I get $Q_2(q)=(1+q)(1-q^4)(1-q^5)$ $=q^{10}+q^9-q^6-2q^5-q^4+q+1$ with two sign changes, not the three that you suggest. Or do you mean $n+1$ same-sign blocks? In either case, the lower-bound side of your statement at least is easy from Descartes... $\endgroup$
    – Steven Stadnicki
    Commented Feb 1, 2022 at 18:55
  • $\begingroup$ @StevenStadnicki: Of yeah, it should be $n$ not $n+1$. $\endgroup$
    – T. Amdeberhan
    Commented Feb 1, 2022 at 18:56
  • $\begingroup$ Quite interesting problem. Numerical results suggests the peak of $|Q_n(q)|$ on the unit circle occurs at about $\pm \pi/(3n+O(1))$, which is consistent with a sign pattern of period $6n+O(1)$ and thus $n+O(1)$ blocks of roughly the same size. The claim is probably true. $\endgroup$
    – Chen Wang
    Commented Nov 6, 2023 at 3:30

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