Abstract : We propose a computational procedure to evaluate the generalized incomplete
gamma function ∫xy sp-1 e-μs ds, for
0 ≤ x < y ≤ +∞, a real number μ ≠ 0 and a positive integer p. Our approach
consists in selecting, according to the value of the parameters x, y, μ, p, the
fastest and most accurate estimate among series expansions, continued fractions,
recursive integration by parts, or, when x ≈ y, a first order trapezoidal
rule. We show that the accuracy reached by our algorithm is nearly optimal for a
large range of parameters.