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<p>In this paper, using Fourier-Legendre series expansions, and performing integral and summation transformations, we present totally 39 double Apéry-type series, which are elegantly expressed in terms of constants such as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1891_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>π</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1891_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(1/\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1891_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ln (2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>ln</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the golden ratio <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1891_Article_IEq4.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>, and the Catalan constant <i>G</i>. Notably, we develop unified approaches to simplify the computational process of these double series by deriving explicit expressions for certain parametric series involving the golden ratio and alternating harmonic numbers.</p>

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Evaluations of some double Apéry-type series via Fourier-Legendre expansions

  • 期刊论文

  • 发表日期：

    2025年5月27日

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  • Ao Chen,
  • Weiping Wang,
  • Jixiang Zhong

摘要

In this paper, using Fourier-Legendre series expansions, and performing integral and summation transformations, we present totally 39 double Apéry-type series, which are elegantly expressed in terms of constants such as \(\pi \) π , \(1/\pi \) 1 / π , \(\ln (2)\) ln ( 2 ) , the golden ratio \(\phi \) ϕ , and the Catalan constant G. Notably, we develop unified approaches to simplify the computational process of these double series by deriving explicit expressions for certain parametric series involving the golden ratio and alternating harmonic numbers.