§33.4 Recurrence Relations and Derivatives

Notes:: See Powell (1947).
Keywords:: Coulomb functions: variables $\rho,\eta$ , derivatives
Referenced by:: §33.23(iv), §33.23(iv)
Permalink:: http://dlmf.nist.gov/33.4

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

33.4.1

$R_{\ell}=\sqrt{1+\dfrac{\eta^{2}}{\ell^{2}}},$

$S_{\ell}=\dfrac{\ell}{\rho}+\dfrac{\eta}{\ell},$

$T_{\ell}=S_{\ell}+S_{{\ell+1}}.$

Symbols:: $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $R_{\ell}$ : factor, $S_{\ell}$ : factor and $T_{\ell}$ : factor
Referenced by:: §33.13, §33.8
Permalink:: http://dlmf.nist.gov/33.4.E1
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Then, with $X_{\ell}$ denoting any of $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ , or $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)$ ,

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
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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

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
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