A note on the recursive calculation of incomplete gamma functions

Abstract

It is known that the recurrence relation for incomplete gamma functions {γ(a + n, x)}, 0 ≤ a < 1, n = 0,1,2, . . . , when x is positive, is unstable - more so the larger x. Nevertheless, the recursion can be used in the range 0 ≤ n ≤ x practically without error growth, and in larger ranges 0 ≤ n ≤ N with a loss of accuracy that can be controlled by suitably limiting N.