The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this paper is to study the degenerate harmonic and degenerate higher-order harmonic numbers, which are respectively degenerate versions of the harmonic and higher-order harmonic numbers, in connection with the degenerate zeta and degenerate Hurwitz zeta function. Here the degenerate zeta and degenerate Hurwitz zeta function are respectively degenerate versions of the Riemann zeta and Hurwitz zeta function. We show that several infinite sums involving the degenerate higher-order harmonic numbers can be expressed in terms of the degenerate zeta function. Furthermore, we demonstrate that an infinite sum involving finite sums of products of the degenerate harmonic numbers can be represented by using the degenerate Hurwitz zeta function.

中文翻译：

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简并谐波和简并高阶谐波数的一些恒等式


谐波数和高阶谐波数经常出现在与组合恒等式相关的几个领域中，许多表达式涉及解析数论中的特殊函数，以及算法分析。本文的目的是研究简并谐波和简并高阶谐波数，它们分别是谐波和高阶谐波数的简并版本，与简并 zeta 和简并 Hurwitz zeta 函数有关。这里，简并 zeta 和简并 Hurwitz zeta 函数分别是 Riemann zeta 和 Hurwitz zeta 函数的简并版本。我们表明，涉及简并高阶谐波数的几个无限和可以用简并 zeta 函数来表示。此外，我们证明了涉及简并谐波数的有限和的无限和可以用简并 Hurwitz zeta 函数来表示。