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33 Coulomb FunctionsVariables \rho,\eta

§33.4 Recurrence Relations and Derivatives

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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}}.
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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),

33.4.2R_{\ell}X_{{\ell-1}}-T_{\ell}X_{{\ell}}+R_{{\ell+1}}X_{{\ell+1}}=0,\ell\geq 1,
33.4.3X_{\ell}^{{\prime}}=R_{\ell}X_{{\ell-1}}-S_{\ell}X_{\ell},\ell\geq 1,
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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:
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33.4.4X_{\ell}^{{\prime}}=S_{{\ell+1}}X_{\ell}-R_{{\ell+1}}X_{{\ell+1}},\ell\geq 0.
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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:
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