On the Hyperharmonic Function

In this paper we investigate some properties of Hyperharmonic function defined$H_{z}^{(w)}=\frac{\left( z\right) _{w}}{z\Gamma\left( w\right) }\left( \Psi\left( z+w\right) -\Psi\left( w\right) \right)$where $\text{ \ \ }w\text{, }z+w\in\mathbb{C}\backslash\left( \mathbb{Z}^{-}\cup\left\{ 0\right\} \right).$ Using this definition we introduce harmonic numbers with complex index and we give some series of these numbers. Also formulas for the calculation of harmonic numbers with rational index are obtained. For the simplicity of differentiation we reorganized representation of $H_{z}^{(w)}$. With the help of this new form we get higher derivatives of Hyperharmonic function more easily. Besides these, owing to the fact that the Hyperharmonic function is composed of some important functions, we interested in properties and connections of it. We get connections between Hyperharmonic function and trigonometric functions. Infinite product representation, integral representation and differentiation identities of this function also obtained.