§13.17 Continued Fractions

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

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

13.17.1 $\frac{\sqrt{z}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)}{\mathop{M_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)}=1+\cfrac{u_{{1}}z}{1+\cfrac{u_{{2}}z}{1+\cdots}},$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : 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

where

13.17.2

$u_{{2n+1}}=-\frac{\frac{1}{2}+\mu+\kappa+n}{(2\mu+2n+1)(2\mu+2n+2)},$

$u_{{2n}}=\frac{\frac{1}{2}+\mu-\kappa+n}{(2\mu+2n)(2\mu+2n+1)}.$

Symbols:: $n$ : nonnegative integer and $u_{n}$ : continued fraction coefficients
Permalink:: http://dlmf.nist.gov/13.17.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

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

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

13.17.3 $\frac{\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)}{\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)}=1+\cfrac{v_{{1}}/z}{1+\cfrac{v_{{2}}/z}{1+\cdots}},$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : 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

where

13.17.4

$v_{{2n+1}}=\tfrac{1}{2}+\mu-\kappa+n,$

$v_{{2n}}=\tfrac{1}{2}-\mu-\kappa+n.$

Symbols:: $n$ : nonnegative integer and $v_{n}$ : continued fraction coefficients
Permalink:: http://dlmf.nist.gov/13.17.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

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

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