  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.
