Schlosser's sign-pattern conjecture for powers of the infinite Borwein product

Let G3(q)G_{3}(q) be the infinite Borwein product

G3(q)=(q;q)(q3;q3).G_{3}(q)=\frac{(q;q)_{\infty}}{(q^{3};q^{3})_{\infty}}.

Suppose that G3(q)δG_{3}(q)^{\delta} has the dissection

G3(q)δ=Aδ(q3)qBδ(q3)q2Cδ(q3).G_{3}(q)^{\delta}=A^{\delta}(q^{3})-qB^{\delta}(q^{3})-q^{2}C^{\delta}(q^{3}).

Schlosser's conjecture. If δ\delta is real and

0.2279981273419732δ1or2δ3,0.227998127341\cdots\approx\frac{9-\sqrt{73}}{2}\leq\delta\leq1\quad\text{or}\quad2\leq\delta\leq3,

then Aδ(q)A^{\delta}(q), Bδ(q)B^{\delta}(q), and Cδ(q)C^{\delta}(q) are power series in qq with non-negative real coefficients. This extends known sign-regularity phenomena for coefficients of infinite products and their powers; the statement is presented as a conjecture, and no resolution is supplied in the source.

Sources & referencesView supporting material

Primary source

Bing He and Linpei Li, “Some conjectures of Schlosser and Zhou on sign patterns of the coefficients of infinite products”, arXiv:2509.10023 (2025).

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