DLMF:33.14.E14 (Q9624)

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DLMF:33.14.E14
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    Defining formula
    ϕ n , ℓ ⁢ ( r ) = ( - 1 ) ℓ + 1 + n ⁢ ( 2 / n 3 ) 1 / 2 ⁢ s ⁡ ( - 1 / n 2 , ℓ ; r ) = ( - 1 ) ℓ + 1 + n n ℓ + 2 ⁢ ( ( n - ℓ - 1 ) ! ( n + ℓ ) ! ) 1 / 2 ⁢ ( 2 ⁢ r ) ℓ + 1 ⁢ e - r / n ⁢ L n - ℓ - 1 ( 2 ⁢ ℓ + 1 ) ⁡ ( 2 ⁢ r / n ) subscript italic-ϕ 𝑛 ℓ 𝑟 superscript 1 ℓ 1 𝑛 superscript 2 superscript 𝑛 3 1 2 Coulomb-s 1 superscript 𝑛 2 ℓ 𝑟 superscript 1 ℓ 1 𝑛 superscript 𝑛 ℓ 2 superscript 𝑛 ℓ 1 𝑛 ℓ 1 2 superscript 2 𝑟 ℓ 1 𝑟 𝑛 Laguerre-polynomial-L 2 ℓ 1 𝑛 ℓ 1 2 𝑟 𝑛 {\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)}}
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    DLMF id
    DLMF:33.14.E14
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