DLMF:18.15.E19 (Q5713)

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DLMF:18.15.E19
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    L n ( α ) ( ν x ) = e 1 2 ν x 2 α x 1 2 α + 1 4 ( 1 - x ) 1 4 ( ξ 1 2 J α ( ν ξ ) m = 0 M - 1 A m ( ξ ) ν 2 m + ξ - 1 2 J α + 1 ( ν ξ ) m = 0 M - 1 B m ( ξ ) ν 2 m + 1 + ξ 1 2 env J α ( ν ξ ) O ( 1 ν 2 M - 1 ) ) , Laguerre-polynomial-L 𝛼 𝑛 𝜈 𝑥 superscript 𝑒 1 2 𝜈 𝑥 superscript 2 𝛼 superscript 𝑥 1 2 𝛼 1 4 superscript 1 𝑥 1 4 superscript 𝜉 1 2 Bessel-J 𝛼 𝜈 𝜉 superscript subscript 𝑚 0 𝑀 1 subscript 𝐴 𝑚 𝜉 superscript 𝜈 2 𝑚 superscript 𝜉 1 2 Bessel-J 𝛼 1 𝜈 𝜉 superscript subscript 𝑚 0 𝑀 1 subscript 𝐵 𝑚 𝜉 superscript 𝜈 2 𝑚 1 superscript 𝜉 1 2 envelope-Bessel-J 𝛼 𝜈 𝜉 Big-O 1 superscript 𝜈 2 𝑀 1 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\nu x\right)=\frac{e^{\frac% {1}{2}\nu x}}{2^{\alpha}x^{\frac{1}{2}\alpha+\frac{1}{4}}(1-x)^{\frac{1}{4}}}% \left(\xi^{\frac{1}{2}}J_{\alpha}\left(\nu\xi\right)\sum_{m=0}^{M-1}\frac{A_{m% }(\xi)}{\nu^{2m}}+\xi^{-\frac{1}{2}}J_{\alpha+1}\left(\nu\xi\right)\sum_{m=0}^% {M-1}\frac{B_{m}(\xi)}{\nu^{2m+1}}+\xi^{\frac{1}{2}}\mathrm{env}\mskip-2.0mu J% _{\alpha}\left(\nu\xi\right)O\left(\frac{1}{\nu^{2M-1}}\right)\right),}}
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    DLMF:18.15.E19
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