# §33.17 Recurrence Relations and Derivatives

 33.17.1 $(\ell+1)r\mathop{f\/}\nolimits\!\left(\epsilon,\ell-1;r\right)-(2\ell+1)\left(% \ell(\ell+1)-r\right)\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)+\ell% \left(1+(\ell+1)^{2}\epsilon\right)r\mathop{f\/}\nolimits\!\left(\epsilon,\ell% +1;r\right)=0,$
 33.17.2 $(\ell+1)\left(1+\ell^{2}\epsilon\right)r\mathop{h\/}\nolimits\!\left(\epsilon,% \ell-1;r\right)-(2\ell+1)\left(\ell(\ell+1)-r\right)\mathop{h\/}\nolimits\!% \left(\epsilon,\ell;r\right)+\ell r\mathop{h\/}\nolimits\!\left(\epsilon,\ell+% 1;r\right)=0,$
 33.17.3 $\displaystyle(\ell+1)r\mathop{f\/}\nolimits'\!\left(\epsilon,\ell;r\right)$ $\displaystyle=\left((\ell+1)^{2}-r\right)\mathop{f\/}\nolimits\!\left(\epsilon% ,\ell;r\right)-{\left(1+(\ell+1)^{2}\epsilon\right)r\mathop{f\/}\nolimits\!% \left(\epsilon,\ell+1;r\right),}$ 33.17.4 $\displaystyle(\ell+1)r\mathop{h\/}\nolimits'\!\left(\epsilon,\ell;r\right)$ $\displaystyle=\left((\ell+1)^{2}-r\right)\mathop{h\/}\nolimits\!\left(\epsilon% ,\ell;r\right)-r\mathop{h\/}\nolimits\!\left(\epsilon,\ell+1;r\right).$