- §8.25(i) Series Expansions
- §8.25(ii) Quadrature
- §8.25(iii) Asymptotic Expansions
- §8.25(iv) Continued Fractions
- §8.25(v) Recurrence Relations

Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $\left|z\right|$ is large owing to slowness of convergence and cancellation. For large $\left|z\right|$ the corresponding asymptotic expansions (generally divergent) are used instead. See also Luke (1975, pp. 101–102) and Temme (1994b).

DiDonato and Morris (1986) describes an algorithm for computing $P\left(a,x\right)$ and $Q\left(a,x\right)$ for $a\ge 0$, $x\ge 0$, and $a+x\ne 0$ from the uniform expansions in §8.12. The algorithm supplies 14S accuracy. A numerical inversion procedure is also given for calculating the value of $x$ (with 10S accuracy), when $a$ and $P\left(a,x\right)$ are specified, based on Newton’s rule (§3.8(ii)). See also Temme (1987, 1994b).

Expansions involving incomplete gamma functions often require the generation of sequences $P\left(a+n,x\right)$, $Q\left(a+n,x\right)$, or ${\gamma}^{*}\left(a+n,x\right)$ for fixed $a$ and $n=0,1,2,\mathrm{\dots}$. An efficient procedure, based partly on the recurrence relations (8.8.5) and (8.8.6), is described in Gautschi (1979b, 1999).