<p>Pertaining to an entry from Ramanujan’s third notebook, this note concerns the special value of the infinite product <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1106_Article_IEq1.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="157" /> </InlineMediaObject> <EquationSource Format="TEX">$\prod _{n=1}^\infty \textrm{e}^{-x} \big (1+x/n^2\big )^{n^2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∏</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <msup> <mtext>e</mtext> <mrow> <mo>-</mo> <mi>x</mi> </mrow> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">/</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> <msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <msup> <mi>n</mi> <mn>2</mn> </msup> </msup> </mrow> </math></EquationSource> </InlineEquation> at <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1106_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">$x=\{(\log \phi ) / \pi \}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mo>log</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>π</mi> <mo stretchy="false">}</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1106_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> denotes the golden ratio. Together with some other infinite products involving the golden ratio, another special value of the same product is briefly considered. Functional equations for products of this type are discussed in connection with Kummer’s work on polylogarithms.</p>

Note on an entry in Ramanujan’s third notebook involving polylogarithms and the golden ratio

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2025年6月9日

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J. G. Bradley-Thrush

摘要

Pertaining to an entry from Ramanujan’s third notebook, this note concerns the special value of the infinite product $\prod _{n=1}^\infty \textrm{e}^{-x} \big (1+x/n^2\big )^{n^2}$ $\prod_{n = 1}^{\infty} e^{- x} (1 + x / n^{2})^{n^{2}}$ at $x=\{(\log \phi ) / \pi \}^2$ $x = {(log ϕ) / π}^{2}$ , where $\phi $ $ϕ$ denotes the golden ratio. Together with some other infinite products involving the golden ratio, another special value of the same product is briefly considered. Functional equations for products of this type are discussed in connection with Kummer’s work on polylogarithms.