- §13.29(i) Series Expansions
- §13.29(ii) Differential Equations
- §13.29(iii) Integral Representations
- §13.29(iv) Recurrence Relations
- §13.29(v) Continued Fractions

Although the Maclaurin series expansion (13.2.2) converges for all finite values of $z$, it is cumbersome to use when $\left|z\right|$ is large owing to slowness of convergence and cancellation. For large $\left|z\right|$ the asymptotic expansions of §13.7 should be used instead. Accuracy is limited by the magnitude of $\left|z\right|$. However, this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied by the combination of (13.7.10) and (13.7.11), or by use of the hyperasymptotic expansions given in Olde Daalhuis and Olver (1995a). For large values of the parameters $a$ and $b$ the approximations in §13.8 are available.

Similarly for the Whittaker functions.

A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation.

For $M\left(a,b,z\right)$ and ${M}_{\kappa ,\mu}\left(z\right)$ this means that in the sector $\left|\mathrm{ph}z\right|\le \pi $ we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2).

For $U\left(a,b,z\right)$ and ${W}_{\kappa ,\mu}\left(z\right)$ we may integrate along outward rays from the origin in the sectors $$, with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). In the sector $$ the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). On the rays $\mathrm{ph}z=\pm \frac{1}{2}\pi $, integration can proceed in either direction.

The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. In Allasia and Besenghi (1991) and Allasia and Besenghi (1987a) the high accuracy of the trapezoidal rule for the computation of Kummer functions is described. Gauss quadrature methods are discussed in Gautschi (2002b).

The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. In the following two examples Olver’s algorithm (§3.6(v)) can be used.

We assume $2\mu \ne -1,-2,-3,\mathrm{\dots}$. Then we have

13.29.1 | $$\frac{{z}^{2}\left(n+\mu -\frac{1}{2}\right)\left({\left(n+\mu +\frac{1}{2}\right)}^{2}-{\kappa}^{2}\right)}{\left(n+\mu \right)\left(n+\mu +\frac{1}{2}\right)\left(n+\mu +1\right)}y\left(n+1\right)+16\left({\left(n+\mu \right)}^{2}-\frac{1}{2}\kappa z-\frac{1}{4}\right)y\left(n\right)-16\left({\left(n+\mu \right)}^{2}-\frac{1}{4}\right)y\left(n-1\right)=0,$$ | ||

with recessive solution

13.29.2 | $$y\left(n\right)={z}^{-n-\mu -\frac{1}{2}}{M}_{\kappa ,n+\mu}\left(z\right),$$ | ||

normalizing relation

13.29.3 | $${\mathrm{e}}^{-\frac{1}{2}z}=\sum _{s=0}^{\mathrm{\infty}}\frac{{\left(2\mu \right)}_{s}{\left(\frac{1}{2}+\mu -\kappa \right)}_{s}}{{\left(2\mu \right)}_{2s}s\mathrm{!}}{\left(-z\right)}^{s}y\left(s\right),$$ | ||

and estimate

13.29.4 | $$y\left(n\right)=1+O\left({n}^{-1}\right),$$ | ||

$n\to \mathrm{\infty}$. | |||

We assume $a,a+1-b\ne 0,-1,-2,\mathrm{\dots}$. Then we have

13.29.5 | $$\left(n+a\right)w\left(n\right)-\left(2\left(n+a+1\right)+z-b\right)w\left(n+1\right)+\left(n+a-b+2\right)w\left(n+2\right)=0,$$ | ||

with recessive solution

13.29.6 | $$w\left(n\right)={\left(a\right)}_{n}U\left(n+a,b,z\right),$$ | ||

normalizing relation

13.29.7 | $${z}^{-a}=\sum _{s=0}^{\mathrm{\infty}}\frac{{\left(a-b+1\right)}_{s}}{s\mathrm{!}}w\left(s\right),$$ | ||

and estimate

13.29.8 | $$w\left(n\right)\sim \frac{\sqrt{\pi}{\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{4}\left(4a-2b+1\right)}}{\mathrm{\Gamma}\left(a\right)\mathrm{\Gamma}\left(a+1-b\right)}{n}^{\frac{1}{4}\left(4a-2b-3\right)}{\mathrm{e}}^{-2\sqrt{nz}},$$ | ||

as $n\to \mathrm{\infty}$. See Temme (1983), and also Wimp (1984, Chapter 5) and Deaño et al. (2010).

In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of $M\left(n,b,x\right)$, when $b$ and $x$ are real and $n$ is a positive integer. The accuracy is controlled and validated by a running error analysis coupled with interval arithmetic.