§8.25 Methods of Computation

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

See Allasia and Besenghi (1987b) for the numerical computation of from (8.6.4) by means of the trapezoidal rule.

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

The computation of and by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). See also Jacobsen et al. (1986) and Temme (1996b, p. 280).

Expansions involving incomplete gamma functions often require the generation of sequences , , or for fixed and . An efficient procedure, based partly on the recurrence relations (8.8.5) and (8.8.6), is described in Gautschi (1979b, 1999).

Stable recursive schemes for the computation of are described in Miller (1960) for and integer . For and real see Amos (1980b) and Chiccoli et al. (1987, 1988). See also Chiccoli et al. (1990a) and Stegun and Zucker (1974).