13 Confluent Hypergeometric Functions Whittaker Functions 13.16 Integral Representations 13.18 Relations to Other Functions

§13.17 Continued Fractions

ⓘ

Keywords:: Whittaker functions, continued fractions
Notes:: See Jones and Thron (1980, Theorems 6.3 and 6.5).
Permalink:: http://dlmf.nist.gov/13.17
See also:: Annotations for Ch.13

If $\kappa,\mu\in\mathbb{C}$ such that $\mu\pm(\kappa-\tfrac{1}{2})\neq-1,-2,-3,\dots$ , then

13.17.1		$\frac{\sqrt{z}M_{\kappa,\mu}\left(z\right)}{M_{\kappa-\frac{1}{2},\mu+\frac{1}% {2}}\left(z\right)}=1+\cfrac{u_{1}z}{1+\cfrac{u_{2}z}{1+\cdots}},$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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	$\displaystyle u_{2n+1}$	$\displaystyle=-\frac{\frac{1}{2}+\mu+\kappa+n}{(2\mu+2n+1)(2\mu+2n+2)},$
13.17.2	$\displaystyle u_{2n}$	$\displaystyle=\frac{\frac{1}{2}+\mu-\kappa+n}{(2\mu+2n)(2\mu+2n+1)}.$
ⓘ 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 $z\in\mathbb{C}$ . For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).

If $\kappa,\mu\in\mathbb{C}$ such that $\mu+\tfrac{1}{2}\pm(\kappa+1)\neq-1,-2,-3,\dots$ , then

13.17.3		$\frac{W_{\kappa,\mu}\left(z\right)}{\sqrt{z}W_{\kappa-\frac{1}{2},\mu-\frac{1}% {2}}\left(z\right)}=1+\cfrac{v_{1}/z}{1+\cfrac{v_{2}/z}{1+\cdots}},$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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	$\displaystyle v_{2n+1}$	$\displaystyle=\tfrac{1}{2}+\mu-\kappa+n,$
13.17.4	$\displaystyle v_{2n}$	$\displaystyle=\tfrac{1}{2}-\mu-\kappa+n.$
ⓘ 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 $|\operatorname{ph}{z}|<\pi$ .

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

13.16 Integral Representations 13.18 Relations to Other Functions

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