In this paper we consider the functional equation for factorial sum and its particular solutions (Kurepa's function K(z) [3] and function K1(z)). We determine an extension of domain of functions K(z) and K1(z) in the sense of Cauchy's principal value at point [2]. In this paper we give an addendum to the proof of Slavic's representation of Kurepa's function K(z) [6]. Also, we consider some representations of functions K(z) and K1(z) via incomplete gamma function and we consider differential transcendency of previous functions too.