<p>The objective of this paper is the analysis of a class of multiple Apéry-like series occurring in quantum electrodynamics by computation of Feynman diagrams. The characteristic aspect of the investigated multiple series consists in the simultaneous inclusion of the binomial-exponential terms <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2963_Article_IEq1.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$\left( {\begin{array}{c}2n\\ n\end{array}}\right) /4^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <mo stretchy="false">/</mo> <msup> <mn>4</mn> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2963_Article_IEq2.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$4^n/\left( {\begin{array}{c}2n\\ n\end{array}}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>4</mn> <mi>n</mi> </msup> <mo stretchy="false">/</mo> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>n</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation> at both the innermost and outermost levels of summation, such that these terms are arranged in a reciprocal configuration.</p>

On a Class of Multiple Apéry-Like Series and Their Reduction

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发表日期：

2025年10月3日

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Marian Genčev,
Pavel Rucki

摘要

The objective of this paper is the analysis of a class of multiple Apéry-like series occurring in quantum electrodynamics by computation of Feynman diagrams. The characteristic aspect of the investigated multiple series consists in the simultaneous inclusion of the binomial-exponential terms $\left( {\begin{array}{c}2n\\ n\end{array}}\right) /4^n$ $(\begin{matrix} 2 n \\ n \end{matrix}) / 4^{n}$ and $4^n/\left( {\begin{array}{c}2n\\ n\end{array}}\right) $ $4^{n} / (\begin{matrix} 2 n \\ n \end{matrix})$ at both the innermost and outermost levels of summation, such that these terms are arranged in a reciprocal configuration.