13 Confluent Hypergeometric Functions Kummer Functions 13.4 Integral Representations 13.6 Relations to Other Functions

§13.5 Continued Fractions

ⓘ

Keywords:: Kummer functions, continued fractions
Notes:: The two continued fractions in this section are Theorems 6.3 and 6.5 in Jones and Thron (1980).
Permalink:: http://dlmf.nist.gov/13.5
See also:: Annotations for Ch.13

If $a,b\in\mathbb{C}$ such that $a\neq-1,-2,-3,\dots$ , and $a-b\neq 0,1,2,\dots$ , then

13.5.1		$\frac{M\left(a,b,z\right)}{M\left(a+1,b+1,z\right)}=1+\cfrac{u_{1}z}{1+\cfrac{% u_{2}z}{1+\cdots}},$
ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, $z$ : complex variable and $u_{n}$ : continued fraction coefficients Referenced by: §13.31(ii) Permalink: http://dlmf.nist.gov/13.5.E1 Encodings: TeX, pMML, png See also: Annotations for §13.5 and Ch.13

where

13.5.2	$\displaystyle u_{2n+1}$	$\displaystyle=\frac{a-b-n}{(b+2n)(b+2n+1)}$ ,
13.5.2	$\displaystyle u_{2n}$	$\displaystyle=\frac{a+n}{(b+2n-1)(b+2n)}$ .
ⓘ Defines: $u_{n}$ : continued fraction coefficients (locally) Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/13.5.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.5 and Ch.13

This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $\mathbb{C}$ . For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).

If $a,b\in\mathbb{C}$ such that $a\neq 0,-1,-2,\dots$ , and $b-a\neq 2,3,4,\dots$ , then

13.5.3		$\frac{U\left(a,b,z\right)}{U\left(a,b-1,z\right)}=1+\cfrac{v_{1}/z}{1+\cfrac{v% _{2}/z}{1+\cdots}},$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $z$ : complex variable and $v_{n}$ : continued fraction coefficients Permalink: http://dlmf.nist.gov/13.5.E3 Encodings: TeX, pMML, png See also: Annotations for §13.5 and Ch.13

where

13.5.4	$\displaystyle v_{2n+1}$	$\displaystyle=a+n$ ,
13.5.4	$\displaystyle v_{2n}$	$\displaystyle=a-b+n+1$ .
ⓘ Defines: $v_{n}$ : continued fraction coefficients (locally) Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/13.5.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.5 and Ch.13

This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector $|\operatorname{ph}{z}|<\pi$ .

See also Cuyt et al. (2008, pp. 322–330).

13.4 Integral Representations 13.6 Relations to Other Functions

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