§13.31 Approximations

Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of and that include the intervals and , respectively, where 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).

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

Let , ,

13.31.1

Symbols:

: generalized hypergeometric function, : Pochhammer’s symbol, : : factorial, : nonnegative integer, : nonnegative integer, : complex variable and : expansion

Permalink:

http://dlmf.nist.gov/13.31.E1

Encodings:

TeX, pMML, png

and

13.31.2

Symbols:

: generalized hypergeometric function, : nonnegative integer, : complex variable and : expansion

Permalink:

http://dlmf.nist.gov/13.31.E2

Encodings:

TeX, pMML, png

Then

13.31.3

Symbols:

: Kummer confluent hypergeometric function, : nonnegative integer, : complex variable, : expansion and : expansion

Permalink:

http://dlmf.nist.gov/13.31.E3

Encodings:

TeX, pMML, png