Evaluation of Incomplete Gamma Functions Using Downward Recursion and Analytical Relations

New downward recursion and analytical approaches are presented for the calculation of incomplete gamma functions (,x) and (,x). The efficiency of calculation with an adequate interval of these expressions is compared with those of other methods and great numerical stability and good rate of convergence is obtained for wide range of integral parameters. It is shown that the downward recursive relations are stable for all values of and x of (,x).