§33.4 Recurrence Relations and Derivatives

For $\mathrm{\ell }=1,2,3,\mathrm{\dots }$, let

33.4.1		$R_{\mathrm{\ell }}$	$=\sqrt{1+{\displaystyle \frac{{\eta }^2}{{\mathrm{\ell }}^2}}},$
		$S_{\mathrm{\ell }}$	$={\displaystyle \frac{\mathrm{\ell }}{\rho }}+{\displaystyle \frac{\eta }{\mathrm{\ell }}},$
		$T_{\mathrm{\ell }}$	$=S_{\mathrm{\ell }}+S_{\mathrm{\ell }+1}.$
ⓘ Defines: $R_{\mathrm{\ell }}$: factor (locally), $S_{\mathrm{\ell }}$: factor (locally) and $T_{\mathrm{\ell }}$: factor (locally) Symbols: $\mathrm{\ell }$: 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 Ch.33

Then, with $X_{\mathrm{\ell }}$ denoting any of $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, or $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$,

33.4.2		$$R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-T_{\mathrm{\ell }}{X}_{\mathrm{\ell }}+R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1}=0,$$
$\mathrm{\ell }\ge 1$,
ⓘ Symbols: $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $T_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: 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 Ch.33

33.4.3		$$X_{\mathrm{\ell }}^{\prime }=R_{\mathrm{\ell }}{X}_{\mathrm{\ell }-1}-S_{\mathrm{\ell }}{X}_{\mathrm{\ell }},$$
$\mathrm{\ell }\ge 1$,
ⓘ Symbols: $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: 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 Ch.33

33.4.4		$$X_{\mathrm{\ell }}^{\prime }=S_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }}-R_{\mathrm{\ell }+1}{X}_{\mathrm{\ell }+1},$$
$\mathrm{\ell }\ge 0$.
ⓘ Symbols: $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $X_{\mathrm{\ell }}$: 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 Ch.33