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.

Incomplete gamma function, asymptotic expansions, recurrence relations
Computation of special functions, construction of tables (msc 65D20)
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