DLMF:18.11.E2 (Q5636)

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DLMF:18.11.E2
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    Statements

    Defining formula
    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).}}
    0 references
    DLMF id
    DLMF:18.11.E2
    0 references
    Symbols used
    $$={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$$ Kummer confluent hypergeometric function
    Defining formula
    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
    0 references
    Kummer confluent hypergeometric function
    Defining formula
    U ⁡ ( a , b , z ) Kummer-confluent-hypergeometric-U 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{b},\NVar{z}\right)}}
    xml-id
    C13.S2.E6.m2adec
    0 references
    Q11333
    Defining formula
    L n ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r12.m2adec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2adec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1adec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1adec
    0 references
     
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