§13.31 Approximations

§13.31(i) Chebyshev-Series Expansions

Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ that include the intervals $0\leq x\leq\alpha$ and $\alpha\leq x<\infty$, respectively, where $\alpha$ is an arbitrary positive constant.

For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985).

§13.31(iii) Rational Approximations

In Luke (1977a) the following rational approximation is given, together with its rate of convergence. For the notation see §16.2(i).

Let $a,a+1-b\neq 0,-1,-2,\dots$, $|\operatorname{ph}z|<\pi$,

 13.31.1 $A_{n}(z)=\sum_{s=0}^{n}\frac{{\left(-n\right)_{s}}{\left(n+1\right)_{s}}{\left% (a\right)_{s}}{\left(b\right)_{s}}}{{\left(a+1\right)_{s}}{\left(b+1\right)_{s% }}(n!)^{2}}\*{{}_{3}F_{3}}\left({-n+s,n+1+s,1\atop 1+s,a+1+s,b+1+s};-z\right),$

and

 13.31.2 $B_{n}(z)={{}_{2}F_{2}}\left({-n,n+1\atop a+1,b+1};-z\right).$

Then

 13.31.3 $z^{a}U\left(a,1+a-b,z\right)=\lim_{n\to\infty}\frac{A_{n}(z)}{B_{n}(z)}.$