DLMF:18.11.E2 (Q5636)

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DLMF:18.11.E2
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    Statements

    test
    L n ( α ) ⁡ ( x ) = ( α + 1 ) n n ! ⁢ M ⁡ ( - n , α + 1 , x ) = ( - 1 ) n n ! ⁢ U ⁡ ( - n , α + 1 , x ) = ( α + 1 ) n n ! ⁢ x - 1 2 ⁢ ( α + 1 ) ⁢ e 1 2 ⁢ x ⁢ M n + 1 2 ⁢ ( α + 1 ) , 1 2 ⁢ α ⁡ ( x ) = ( - 1 ) n n ! ⁢ x - 1 2 ⁢ ( α + 1 ) ⁢ e 1 2 ⁢ x ⁢ W n + 1 2 ⁢ ( α + 1 ) , 1 2 ⁢ α ⁡ ( x ) . Laguerre-polynomial-L 𝛼 𝑛 𝑥 Pochhammer 𝛼 1 𝑛 𝑛 Kummer-confluent-hypergeometric-M 𝑛 𝛼 1 𝑥 superscript 1 𝑛 𝑛 Kummer-confluent-hypergeometric-U 𝑛 𝛼 1 𝑥 Pochhammer 𝛼 1 𝑛 𝑛 superscript 𝑥 1 2 𝛼 1 superscript 𝑒 1 2 𝑥 Whittaker-confluent-hypergeometric-M 𝑛 1 2 𝛼 1 1 2 𝛼 𝑥 superscript 1 𝑛 𝑛 superscript 𝑥 1 2 𝛼 1 superscript 𝑒 1 2 𝑥 Whittaker-confluent-hypergeometric-W 𝑛 1 2 𝛼 1 1 2 𝛼 𝑥 {\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x\right)=\frac{{\left(% \alpha+1\right)_{n}}}{n!}M\left(-n,\alpha+1,x\right)=\frac{(-1)^{n}}{n!}U\left% (-n,\alpha+1,x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}x^{-\frac{1}{2}(% \alpha+1)}e^{\frac{1}{2}x}M_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x% \right)=\frac{(-1)^{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}W_{n+% \frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right).}}
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    DLMF id
    DLMF:18.11.E2
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    Symbols used
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    test
    M ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E2.m2adec
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