Some considerations in connection with alternating Kurepa's function

In this paper we consider the functional equation for alternating factorial sum and some of its particular solutions (alternating Kurepa's function $A(z)$ from [Petojevic_02] and function $A_{1}(z)$). We determine an extension of domain of functions $A(z)$ and $A_{1}(z)$ in the sense of the principal value at point [Slavic_73], [Mijajlovic & Malesevic_07]. Using the methods from [Slavic_73] and [Malesevic_03] we give a new representation of alternating Kurepa's function $A(z)$, which is an analog of Slavi\' c's representation of Kurepa's function $K(z)$ [Slavic_73], [Marichev_83]. Also, we consider some representations of functions $A(z)$ and $A_{1}(z)$ via incomplete gamma function and we consider differential transcendency of previous functions too.