§33.6 Power-Series Expansions in $\rho$

33.6.1		$$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}{A}_k^{\mathrm{\ell }}(\eta ){\rho }^k,$$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: 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 See also: Annotations for §33.6 and Ch.33

33.6.2		$$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}k{A}_k^{\mathrm{\ell }}(\eta ){\rho }^{k-1},$$
ⓘ Symbols: $C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: 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 See also: Annotations for §33.6 and Ch.33

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

33.6.3		$$(k+\mathrm{\ell })(k-\mathrm{\ell }-1)A_k^{\mathrm{\ell }}=2\eta A_{k-1}^{\mathrm{\ell }}-A_{k-2}^{\mathrm{\ell }},$$
$k=\mathrm{\ell }+3,\mathrm{\ell }+4,\mathrm{\dots }$,
ⓘ Symbols: $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient A&S Ref: 14.1.6 Permalink: http://dlmf.nist.gov/33.6.E3 Encodings: TeX, pMML, png See also: Annotations for §33.6 and Ch.33

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

33.6.4		$$A_k^{\mathrm{\ell }}(\eta )=\frac{{(-\mathrm{i})}^{k-\mathrm{\ell }-1}}{(k-\mathrm{\ell }-1)!}{}_{2}F_1(\mathrm{\ell }+1-k,\mathrm{\ell }+1-\mathrm{i}\eta ;2\mathrm{\ell }+2;2).$$
ⓘ Symbols: ${}_{2}F_1(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient Referenced by: §33.6 Permalink: http://dlmf.nist.gov/33.6.E4 Encodings: TeX, pMML, png See also: Annotations for §33.6 and Ch.33

33.6.5		$$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )=\begin{array}{l}\frac{{\mathrm{e}}^{\pm \mathrm{i}{\theta }_{\mathrm{\ell }}(\eta ,\rho )}}{(2\mathrm{\ell }+1)!\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)}\\ \times (\begin{array}{l}\sum _{k=0}^{\mathrm{\infty }}\begin{array}{l}\frac{{\left(a\right)}_k}{{\left(2\mathrm{\ell }+2\right)}_{k}k!}{(\mp 2\mathrm{i}\rho )}^{a+k}\\ \times \left(\ln \left(\mp 2\mathrm{i}\rho \right)+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(2\mathrm{\ell }+2+k\right)\right)\end{array}\\ -\sum _{k=1}^{2\mathrm{\ell }+1}\frac{(2\mathrm{\ell }+1)!(k-1)!}{(2\mathrm{\ell }+1-k)!{\left(1-a\right)}_k}{(\mp 2\mathrm{i}\rho )}^{a-k}),\end{array}\end{array}$$
ⓘ Symbols: ${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $a$: variable A&S Ref: 14.1.14 (equivalent) Referenced by: §33.13, §33.6, §33.6, Erratum (V1.0.19) for Equation (33.6.5) Permalink: http://dlmf.nist.gov/33.6.E5 Encodings: TeX, pMML, png Errata (effective with 1.0.19): On the right-hand side the factor in the denominator was incorrectly written as $\mathrm{\Gamma }\left(-\mathrm{\ell }+\mathrm{i}\eta \right)$. This has been corrected to $\mathrm{\Gamma }\left(-\mathrm{\ell }\pm \mathrm{i}\eta \right)$. Reported 2018-05-15 by Ian Thompson See also: Annotations for §33.6 and Ch.33

where $a=1+\mathrm{\ell }\pm \mathrm{i}\eta$ and $\psi \left(x\right)={\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$ (§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 $H_{\mathrm{\ell }}^{\pm }{}^{\prime }(\eta ,\rho )$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).