<p>The signs of Fourier coefficients of certain eta quotients are determined by dissecting expansions for theta functions and by applying a general dissection formula for certain classes of quintuple products. A characterization is given for the coefficient sign patterns for <Equation ID="Equ15"> <EquationSource Format="TEX">$\begin{aligned} \frac{(q^i;q^i)_{\infty }}{(q^p;q^p)_{\infty }} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfrac> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mi>i</mi> </msup> <mo>;</mo> <msup> <mi>q</mi> <mi>i</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mi>p</mi> </msup> <mo>;</mo> <msup> <mi>q</mi> <mi>p</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> </mfrac> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ i > 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and primes <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ p > 3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. The sign analysis for this quotient addresses and extends a conjecture of Bringmann et al. for the coefficients of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ (q^2;q^2)_{\infty }(q^5;q^5)_{\infty }^{-1} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>∞</mi> </msub> <msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>5</mn> </msup> <mo>;</mo> <msup> <mi>q</mi> <mn>5</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>∞</mi> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation>. The sign distribution for additional classes of eta quotients is considered. This addresses multiple conjectures posed by Bringmann et al.</p>

Sign Patterns of Certain Infinite Products

期刊论文
发表日期：

2025年11月27日

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Zeyu Huang,
Timothy Huber,
James McLaughlin,
Pengjun Wang,
Yan Xu,
Dongxi Ye

摘要

The signs of Fourier coefficients of certain eta quotients are determined by dissecting expansions for theta functions and by applying a general dissection formula for certain classes of quintuple products. A characterization is given for the coefficient sign patterns for $\begin{aligned} \frac{(q^i;q^i)_{\infty }}{(q^p;q^p)_{\infty }} \end{aligned}$ $\begin{matrix} \frac{{(q^{i}; q^{i})}_{\infty}}{{(q^{p}; q^{p})}_{\infty}} \end{matrix}$ for integers $ i > 1 $ $i > 1$ and primes $ p > 3 $ $p > 3$ . The sign analysis for this quotient addresses and extends a conjecture of Bringmann et al. for the coefficients of $ (q^2;q^2)_{\infty }(q^5;q^5)_{\infty }^{-1} $ ${(q^{2}; q^{2})}_{\infty} {(q^{5}; q^{5})}_{\infty}^{- 1}$ . The sign distribution for additional classes of eta quotients is considered. This addresses multiple conjectures posed by Bringmann et al.