An algorithm for computing the incomplete gamma function γ∗(a, z) for real values of the parameter a and negative real values of the argument z is presented. The algorithm combines the use of series expansions, Poincare-type expansions, uniform asymptotic expansions, and recurrence relations, depending on the parameter region. A relative accuracy ∼10-13 in the parameter region (a, z) [-500, 500] × [-500, 0) can be obtained when computing the function γ∗(a, z) with the Fortran 90 module IncgamNEG implementing the algorithm.

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doi.org/10.1145/2972951
ACM Transactions on Mathematical Software

Gil, A, Ruiz-Antolin, D, Segura, J, & Temme, N.M. (2016). Algorithm 969: computation of the incomplete gamma function for negative values of the argument. ACM Transactions on Mathematical Software, 43(3). doi:10.1145/2972951