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§14.32 Methods of Computation

Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters μ and ν the power-series expansions of the various hypergeometric function representations given in §§14.3(i)14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967).

Other methods include:

  • Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)14.20(ix).

  • Numerical integration (§3.7) of the defining differential equations (14.2.2), (14.20.1), and (14.21.1).

  • Quadrature (§3.5) of the integral representations given in §§14.12, 14.19(iii), 14.20(iv), and 14.25; see Segura and Gil (1999) and Gil et al. (2000).

  • Evaluation (§3.10) of the continued fractions given in §14.14. See Gil and Segura (2000).

  • For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).