DLMF:13.14.E9 (Q4498)

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DLMF:13.14.E9
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    Statements

    Defining formula
    W κ , ± 1 2 ⁢ n ⁡ ( z ) = ( - 1 ) κ - 1 2 ⁢ n - 1 2 ⁢ e - 1 2 ⁢ z ⁢ z 1 2 ⁢ n + 1 2 ⁢ ∑ k = 0 κ - 1 2 ⁢ n - 1 2 ( κ - 1 2 ⁢ n - 1 2 k ) ⁢ ( n + 1 + k ) κ - k - 1 2 ⁢ n - 1 2 ⁢ ( - z ) k , Whittaker-confluent-hypergeometric-W 𝜅 plus-or-minus 1 2 𝑛 𝑧 superscript 1 𝜅 1 2 𝑛 1 2 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝑛 1 2 superscript subscript 𝑘 0 𝜅 1 2 𝑛 1 2 binomial 𝜅 1 2 𝑛 1 2 𝑘 Pochhammer 𝑛 1 𝑘 𝜅 𝑘 1 2 𝑛 1 2 superscript 𝑧 𝑘 {\displaystyle{\displaystyle W_{\kappa,\pm\frac{1}{2}n}\left(z\right)=(-1)^{% \kappa-\frac{1}{2}n-\frac{1}{2}}e^{-\frac{1}{2}z}z^{\frac{1}{2}n+\frac{1}{2}}% \sum_{k=0}^{\kappa-\frac{1}{2}n-\frac{1}{2}}\genfrac{(}{)}{0.0pt}{}{\kappa-% \frac{1}{2}n-\frac{1}{2}}{k}{\left(n+1+k\right)_{\kappa-k-\frac{1}{2}n-\frac{1% }{2}}}(-z)^{k},}}
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    DLMF id
    DLMF:13.14.E9
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    constraint
    κ - 1 2 ⁢ n - 1 2 = 0 , 1 , 2 , … 𝜅 1 2 𝑛 1 2 0 1 2 … {\displaystyle{\displaystyle\kappa-\frac{1}{2}n-\frac{1}{2}=0,1,2,\dots}}
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    Symbols used
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
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    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.m2acdec
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    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2agdec
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    Q11559
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1cdec
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    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1hdec
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