§13.31 Approximations

Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.31

§13.31(i) Chebyshev-Series Expansions
§13.31(ii) Padé Approximations
§13.31(iii) Rational Approximations

§13.31(i) Chebyshev-Series Expansions

Keywords:: Chebyshev-series expansions, Kummer functions
Permalink:: http://dlmf.nist.gov/13.31.i

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

§13.31(ii) Padé Approximations

Permalink:: http://dlmf.nist.gov/13.31.ii

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

Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.31.iii

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

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

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