The Generalized Harmonic Numbers with Parameters $\delta$ and $\gamma$, $H_{l,\kappa}(\delta,\gamma)$, and Their Applications

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.