Some considerations in connection with Kurepa's function

In this paper we consider the functional equation for factorial sum and its particular solutions (Kurepa's function $K(z)$ \cite{Kurepa_71} and function $K_{1}(z)$). We determine an extension of domain of functions $K(z)$ and $K_{1}(z)$ in the sense of Cauchy's principal value at point \cite{Slavic_70}. In this paper we give an addendum to the proof of Slavi\' c's representation of Kurepa's function $K(z)$ \cite{Slavic_73}. Also, we consider some representations of functions $K(z)$ and $K_{1}(z)$ via incomplete gamma function and we consider differential transcendency of previous functions too.