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