§13.17 Continued Fractions

If $\kappa ,\mu \in ℂ$ such that $\mu \pm (\kappa -\frac{1}{2})\ne -1,-2,-3,\mathrm{\dots }$, then

13.17.1		$$\frac{\sqrt{z}{M}_{\kappa ,\mu }\left(z\right)}{M_{\kappa -\frac{1}{2},\mu +\frac{1}{2}}\left(z\right)}=1+\frac{u_{1}z}{1+}\frac{u_{2}z}{1+}\mathrm{\cdots },$$
ⓘ Symbols: $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $u_n$: continued fraction coefficients Permalink: http://dlmf.nist.gov/13.17.E1 Encodings: TeX, pMML, png See also: Annotations for §13.17 and Ch.13

where

13.17.2		$u_{2n+1}$	$=-{\displaystyle \frac{\frac{1}{2}+\mu +\kappa +n}{(2\mu +2n+1)(2\mu +2n+2)}},$
		$u_{2n}$	$={\displaystyle \frac{\frac{1}{2}+\mu -\kappa +n}{(2\mu +2n)(2\mu +2n+1)}}.$
ⓘ Defines: $u_n$: continued fraction coefficients (locally) Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/13.17.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.17 and Ch.13

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

If $\kappa ,\mu \in ℂ$ such that $\mu +\frac{1}{2}\pm (\kappa +1)\ne -1,-2,-3,\mathrm{\dots }$, then

13.17.3		$$\frac{W_{\kappa ,\mu }\left(z\right)}{\sqrt{z}{W}_{\kappa -\frac{1}{2},\mu -\frac{1}{2}}\left(z\right)}=1+\frac{v_1/z}{1+}\frac{v_2/z}{1+}\mathrm{\cdots },$$
ⓘ Symbols: $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $v_n$: continued fraction coefficients Permalink: http://dlmf.nist.gov/13.17.E3 Encodings: TeX, pMML, png See also: Annotations for §13.17 and Ch.13

where

13.17.4		$v_{2n+1}$	$=\frac{1}{2}+\mu -\kappa +n,$
		$v_{2n}$	$=\frac{1}{2}-\mu -\kappa +n.$
ⓘ Defines: $v_n$: continued fraction coefficients (locally) Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/13.17.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §13.17 and Ch.13

This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector .

See also Cuyt et al. (2008, pp. 336–337).