Some remarks on incomplete gamma type function γ*(α, x_)

The incomplete gamma type function γ*(α, x_) is defined as locally summable function on the real line for α>0 by γ*(α,x_) = {∫x0 |u|α-1 e-u du, x≤0; 0, x > 0 = ∫-x_0 |u|α-1 e-u du the integral divergining α ≤ 0 and by using the recurrence relation γ*(α + 1,x_) = -αγ*(α,x_) - xα_ e-x the definition of γ*(α, x_) can be extended to the negative non-integer values of α. Recently the authors [8] defined γ*(-m, x_) for m = 0, 1, 2,... . In this paper we define the derivatives of the incomplete gamma type function γ*(α, x_) as a distribution for all α < 0.