In this note, we study the infinite version of the Mneimneh sum, which involves binomial coefficients and harmonic numbers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\sum }_{k=1}^{\infty }\left(\begin{array}{c}\alpha \\ k\end{array}\right){p}^{k}{\left(1-p\right)}^{\alpha -k}{H}_{k}\left(z\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mi>α</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>k</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> <msup> <mrow> <mi>p</mi> </mrow> <mi>k</mi> </msup> <msup> <mrow> <mfenced close=")" open="("> <mn>1</mn> <mo>-</mo> <mi>p</mi> </mfenced> </mrow> <mrow> <mi>α</mi> <mo>-</mo> <mi>k</mi> </mrow> </msup> <msub> <mi>H</mi> <mi>k</mi> </msub> <mfenced close=")" open="("> <mi>z</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, where α ∈ ℝ is not a negative integer, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${H}_{k}\left(z\right)={\sum }_{j=1}^{k}{z}^{j}/j$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>H</mi> <mi>k</mi> </msub> <mfenced close=")" open="("> <mi>z</mi> </mfenced> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <msup> <mrow> <mi>z</mi> </mrow> <mi>j</mi> </msup> <mo stretchy="false">/</mo> <mi>j</mi> </mrow> </math></EquationSource> </InlineEquation> is the parameterized analogue of the kth harmonic number of order one. For z = 1 and positive integers α, these binomial sums were investigated long time ago. In this note, we present an infinite version of Mneimneh’s summation formula and establish several new identities by using the well-known integral representation of a parameterized analogue of harmonic numbers.

Infinite version of Mneimneh’s binomial sum of harmonic numbers

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发表日期：

2025年10月1日

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Ende Pan,
Ce Xu

摘要

In this note, we study the infinite version of the Mneimneh sum, which involves binomial coefficients and harmonic numbers ${\sum }_{k=1}^{\infty }\left(\begin{array}{c}\alpha \\ k\end{array}\right){p}^{k}{\left(1-p\right)}^{\alpha -k}{H}_{k}\left(z\right)$ $\sum_{k = 1}^{\infty} (\begin{matrix} α \\ k \end{matrix}) p^{k} {(1 - p)}^{α - k} H_{k} (z)$ , where α ∈ ℝ is not a negative integer, and ${H}_{k}\left(z\right)={\sum }_{j=1}^{k}{z}^{j}/j$ $H_{k} (z) = \sum_{j = 1}^{k} z^{j} / j$ is the parameterized analogue of the kth harmonic number of order one. For z = 1 and positive integers α, these binomial sums were investigated long time ago. In this note, we present an infinite version of Mneimneh’s summation formula and establish several new identities by using the well-known integral representation of a parameterized analogue of harmonic numbers.