§33.6 Power-Series Expansions in $\rho$

Notes:: For (33.6.5) use the definition (33.2.8) with $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ expanded as in (13.2.9). For (33.6.4) use (33.2.4) with Eq. (1.12) of Buchholz (1969).
Keywords:: Coulomb functions: variables $\rho,\eta$
Referenced by:: §33.23(ii), §33.23(iii), §33.23(iii)
Permalink:: http://dlmf.nist.gov/33.6

33.6.1 $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)\sum _{{k=\ell+1}}^{{\infty}}A_{k}^{\ell}(\eta)\rho^{k},$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $A_{{\ell+1}}^{{\ell}}$ : coefficient
A&S Ref:: 14.1.4, 14.1.5
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E1
Encodings:: TeX, pMML, png

33.6.2 ${\mathop{F_{{\ell}}\/}\nolimits^{{\prime}}}\!\left(\eta,\rho\right)=\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)\sum _{{k=\ell+1}}^{{\infty}}kA_{k}^{\ell}(\eta)\rho^{{k-1}},$

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $A_{{\ell+1}}^{{\ell}}$ : coefficient
A&S Ref:: 14.1.4, 14.1.5
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E2
Encodings:: TeX, pMML, png

where $A_{{\ell+1}}^{\ell}=1$ , $A_{{\ell+2}}^{\ell}=\eta/(\ell+1)$ , and

33.6.3 $(k+\ell)(k-\ell-1)A_{k}^{\ell}=2\eta A_{{k-1}}^{\ell}-A_{{k-2}}^{\ell},$ $k=\ell+3,\ell+4,\dots$ ,

Symbols:: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $A_{{\ell+1}}^{{\ell}}$ : coefficient
A&S Ref:: 14.1.6
Permalink:: http://dlmf.nist.gov/33.6.E3
Encodings:: TeX, pMML, png

or in terms of the hypergeometric function (§§15.1, 15.2(i)),

33.6.4 $A_{k}^{\ell}(\eta)=\dfrac{(-i)^{{k-\ell-1}}}{(k-\ell-1)!}\*\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(\ell+1-k,\ell+1-i\eta;2\ell+2;2\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $!$ : $n!$ : factorial, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $A_{{\ell+1}}^{{\ell}}$ : coefficient
Referenced by:: §33.6
Permalink:: http://dlmf.nist.gov/33.6.E4
Encodings:: TeX, pMML, png

33.6.5 $\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)=\frac{e^{{\pm i\mathop{{\theta _{{\ell}}}\/}\nolimits\!\left(\eta,\rho\right)}}}{(2\ell+1)!\mathop{\Gamma\/}\nolimits\!\left(-\ell+i\eta\right)}\left(\sum _{{k=0}}^{\infty}\frac{\left(a\right)_{{k}}}{\left(2\ell+2\right)_{{k}}k!}(\mp 2i\rho)^{{a+k}}\left(\mathop{\ln\/}\nolimits\!\left(\mp 2i\rho\right)+\mathop{\psi\/}\nolimits\!\left(a+k\right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!\left(2\ell+2+k\right)\right)-\sum _{{k=1}}^{{2\ell+1}}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)!\left(1-a\right)_{{k}}}(\mp 2i\rho)^{{a-k}}\right),$

where $a=1+\ell\pm i\eta$ and $\mathop{\psi\/}\nolimits\!\left(x\right)={\mathop{\Gamma\/}\nolimits^{{\prime}}}\!\left(x\right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$ (§5.2(i)).

The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of $\rho$ . Corresponding expansions for ${\mathop{{H^{{\pm}}_{{\ell}}}\/}\nolimits^{{\prime}}}\!\left(\eta,\rho\right)$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).