# §13.17 Continued Fractions

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

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

where

 13.17.2 $\displaystyle u_{2n+1}$ $\displaystyle=-\frac{\frac{1}{2}+\mu+\kappa+n}{(2\mu+2n+1)(2\mu+2n+2)},$ $\displaystyle u_{2n}$ $\displaystyle=\frac{\frac{1}{2}+\mu-\kappa+n}{(2\mu+2n)(2\mu+2n+1)}.$ 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 $\frac{\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)}{\sqrt{z}\mathop{W_{% \kappa-\frac{1}{2},\mu-\frac{1}{2}}\/}\nolimits\!\left(z\right)}=1+\cfrac{v_{1% }/z}{1+\cfrac{v_{2}/z}{1+\cdots}},$

where

 13.17.4 $\displaystyle v_{2n+1}$ $\displaystyle=\tfrac{1}{2}+\mu-\kappa+n,$ $\displaystyle v_{2n}$ $\displaystyle=\tfrac{1}{2}-\mu-\kappa+n.$ 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 $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\pi$.