Formula:10316

Digital Library of Mathematical Functions ID 33.14.E14

${\displaystyle{\displaystyle\phi_{n,\ell}(r)=(-1)^{\ell+1+n}(2/n^{3})^{1/2}s\left(-1/n^{2},\ell;r\right)=\frac{(-1)^{\ell+1+n}}{n^{\ell+2}}\left(\frac{(n-\ell-1)!}{(n+\ell)!}\right)^{1/2}(2r)^{\ell+1}{\mathrm{e}^{-r/n}}L^{(2\ell+1)}_{n-\ell-1}\left(2r/n\right)}}$

Symbols List

• ${\displaystyle{\displaystyle s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)}}$: regular Coulomb function
• ${\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Laguerre (or generalized Laguerre) polynomial
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\ell}}$: nonnegative integer
• ${\displaystyle{\displaystyle r}}$: real variable
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