In this paper we study the function \[ K_i(z)=\frac1{(i-1)!}ıt^ıfty_0e^{-x}x^{i-1}\frac{x^z-1}{x-1}dx\quad(peratorname{Re}(z)>0, iı\mathbb N) \] defined in [13]. We give the generating functions, some representation and the congruences of the function $K_i(z)$. Also, we present some inequalities for the function $K_i(x)$ for positive values of $x$.