In particular, with this approach, we have improved the understanding of their structural properties and revealed new insights for further research.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we define generalized harmonic numbers with parameters $\delta $ and $\gamma $, $H_{l,\kappa }\left( \delta ,\gamma \right) .$ As their implementation, we obtain some results and combinatorial identities by using the partial derivative and generating functions.
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Key Findings & Data:Additionally, we give graphs of the roots of the equations $H_{l,\kappa }\left( \delta ,\gamma \right)=0 $, $G_{l}\left( \delta ,\gamma \right)=0$ and $G_{l}^{(\kappa )}\left( \delta ,\gamma \right)=0 $ for different parameters.
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Conclusion & Impact:In particular, with this approach, we have improved the understanding of their structural properties and revealed new insights for further research.
In this paper, we define generalized harmonic numbers with parameters $\delta $ and $\gamma $, $H_{l,\kappa }\left( \delta ,\gamma \right) .$ As their implementation, we obtain some results and combinatorial identities by using the partial derivative and generating functions. Additionally, we give graphs of the roots of the equations $H_{l,\kappa }\left( \delta ,\gamma \right)=0 $, $G_{l}\left( \delta ,\gamma \right)=0$ and $G_{l}^{(\kappa )}\left( \delta ,\gamma \right)=0 $ for different parameters. Thus, we have made the motivation and roots of the given equations more understandable. In particular, with this approach, we have improved the understanding of their structural properties and revealed new insights for further research.
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