# §33.4 Recurrence Relations and Derivatives

For $\ell=1,2,3,\dots$, let

 33.4.1 $\displaystyle R_{\ell}$ $\displaystyle=\sqrt{1+\dfrac{\eta^{2}}{\ell^{2}}},$ $\displaystyle S_{\ell}$ $\displaystyle=\dfrac{\ell}{\rho}+\dfrac{\eta}{\ell},$ $\displaystyle T_{\ell}$ $\displaystyle=S_{\ell}+S_{\ell+1}.$ ⓘ Defines: $R_{\ell}$: factor (locally), $S_{\ell}$: factor (locally) and $T_{\ell}$: factor (locally) Symbols: $\ell$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter Referenced by: §33.13, §33.8 Permalink: http://dlmf.nist.gov/33.4.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 33.4 and 33

Then, with $X_{\ell}$ denoting any of $F_{\ell}\left(\eta,\rho\right)$, $G_{\ell}\left(\eta,\rho\right)$, or ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$,

 33.4.2 $R_{\ell}X_{\ell-1}-T_{\ell}X_{\ell}+R_{\ell+1}X_{\ell+1}=0,$ $\ell\geq 1$, ⓘ Symbols: $\ell$: nonnegative integer, $R_{\ell}$: factor, $T_{\ell}$: factor and $X_{\ell}$: any Coulomb function A&S Ref: 14.2.3 Permalink: http://dlmf.nist.gov/33.4.E2 Encodings: TeX, pMML, png See also: Annotations for 33.4 and 33
 33.4.3 $X_{\ell}^{\prime}=R_{\ell}X_{\ell-1}-S_{\ell}X_{\ell},$ $\ell\geq 1$, ⓘ Symbols: $\ell$: nonnegative integer, $R_{\ell}$: factor, $S_{\ell}$: factor and $X_{\ell}$: any Coulomb function A&S Ref: 14.2.1 Referenced by: §33.6 Permalink: http://dlmf.nist.gov/33.4.E3 Encodings: TeX, pMML, png See also: Annotations for 33.4 and 33
 33.4.4 $X_{\ell}^{\prime}=S_{\ell+1}X_{\ell}-R_{\ell+1}X_{\ell+1},$ $\ell\geq 0$. ⓘ Symbols: $\ell$: nonnegative integer, $R_{\ell}$: factor, $S_{\ell}$: factor and $X_{\ell}$: any Coulomb function A&S Ref: 14.2.2 Referenced by: §33.6 Permalink: http://dlmf.nist.gov/33.4.E4 Encodings: TeX, pMML, png See also: Annotations for 33.4 and 33