# §14.14 Continued Fractions

 14.14.1 $\tfrac{1}{2}\left(x^{2}-1\right)^{1/2}\frac{\mathop{P^{\mu}_{\nu}\/}\nolimits% \!\left(x\right)}{\mathop{P^{\mu-1}_{\nu}\/}\nolimits\!\left(x\right)}=\cfrac{% x_{0}}{y_{0}+\cfrac{x_{1}}{y_{1}+\cfrac{x_{2}}{y_{2}+\cdots}}},$

where

 14.14.2 $\displaystyle x_{k}$ $\displaystyle=\tfrac{1}{4}(\nu-\mu-k+1)(\nu+\mu+k)\left(x^{2}-1\right),$ $\displaystyle y_{k}$ $\displaystyle=(\mu+k)x,$

provided that $x_{k+1}$ and $y_{k}$ do not vanish simultaneously for any $k=0,1,2,\dots$.

 14.14.3 $(\nu-\mu)\frac{\mathop{Q^{\mu}_{\nu}\/}\nolimits\!\left(x\right)}{\mathop{Q^{% \mu}_{\nu-1}\/}\nolimits\!\left(x\right)}=\cfrac{x_{0}}{y_{0}-\cfrac{x_{1}}{y_% {1}-\cfrac{x_{2}}{y_{2}-\cdots}}},$ $\nu\neq\mu$,

where now

 14.14.4 $\displaystyle x_{k}$ $\displaystyle=(\nu+\mu+k)(\nu-\mu+k),$ $\displaystyle y_{k}$ $\displaystyle=(2\nu+2k+1)x,$

again provided $x_{k+1}$ and $y_{k}$ do not vanish simultaneously for any $k=0,1,2,\dots$.