<p>We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as <Equation ID="Equ43"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_995_Article_Equ43.gif" Format="GIF" Height="57" Rendition="HTML" Resolution="72" Type="Linedraw" Width="442" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \int _0^{e^{-2\pi /\sqrt{10}}} q\prod _{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \textrm{d}q = \frac{1}{4}\left( \sqrt{10-4\sqrt{5}}-1\right) . \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">/</mo> <msqrt> <mn>10</mn> </msqrt> </mrow> </msup> </msubsup> <mi>q</mi> <munderover> <mo>∏</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>q</mi> <mi>j</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>q</mi> <mrow> <mn>10</mn> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mn>8</mn> </msup> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>q</mi> <mrow> <mn>5</mn> <mi>j</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mn>7</mn> </msup> </mfrac> <mtext>d</mtext> <mi>q</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mfenced close=")" open="("> <msqrt> <mrow> <mn>10</mn> <mo>-</mo> <mn>4</mn> <msqrt> <mn>5</mn> </msqrt> </mrow> </msqrt> <mo>-</mo> <mn>1</mn> </mfenced> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.</p>

Ramanujan–Fine integrals for level 10

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发表日期：

2025年1月28日

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Shaun Cooper,
Timothy Huber,
Jeffery Opoku

摘要

We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as $\begin{aligned} \int _0^{e^{-2\pi /\sqrt{10}}} q\prod _{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \textrm{d}q = \frac{1}{4}\left( \sqrt{10-4\sqrt{5}}-1\right) . \end{aligned}$ $\begin{matrix} \int_{0}^{e^{- 2 π / \sqrt{10}}} q \prod_{j = 1}^{\infty} \frac{{(1 - q^{j})}^{3} {(1 - q^{10 j})}^{8}}{{(1 - q^{5 j})}^{7}} d q = \frac{1}{4} (\sqrt{10 - 4 \sqrt{5}} - 1) . \end{matrix}$ We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.