On defining the incomplete gamma function - Fisher, Brian; Jolevsaka-Tuneska, Biljana; KiliÇman, Adem (Taylor and Francis Group, 2003)
The incomplete Gamma function γ(α, x+) is defined as locally summable function on the real line for α > 0 by γ(α,x+)= ∫0x+ uα-1e-udu, the integral diverging for α ≤ 0. The incomplete Gamma function can be defined as a distribution for α< 0 and α ≠ -1, - 2,... by using the recurrence formula γ(α + 1, x+) = αγ(α x+) - x+αe-x. In the following, we define the distribution γ(-m, x+) for m = 0, 1, 2, ....