DLMF:13.14.E8 (Q4497)

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DLMF:13.14.E8
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    Statements

    Defining formula
    W κ , ± 1 2 ⁢ n ⁡ ( z ) = ( - 1 ) n ⁢ e - 1 2 ⁢ z ⁢ z 1 2 ⁢ n + 1 2 n ! ⁢ Γ ⁡ ( 1 2 - 1 2 ⁢ n - κ ) ⁢ ( ∑ k = 1 n n ! ⁢ ( k - 1 ) ! ( n - k ) ! ⁢ ( κ + 1 2 - 1 2 ⁢ n ) k ⁢ z - k - ∑ k = 0 ∞ ( 1 2 ⁢ n + 1 2 - κ ) k ( n + 1 ) k ⁢ k ! ⁢ z k ⁢ ( ln ⁡ z + ψ ⁡ ( 1 2 ⁢ n + 1 2 - κ + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + 1 + k ) ) ) , Whittaker-confluent-hypergeometric-W 𝜅 plus-or-minus 1 2 𝑛 𝑧 superscript 1 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 1 2 𝑛 1 2 𝑛 Euler-Gamma 1 2 1 2 𝑛 𝜅 superscript subscript 𝑘 1 𝑛 𝑛 𝑘 1 𝑛 𝑘 Pochhammer 𝜅 1 2 1 2 𝑛 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 Pochhammer 1 2 𝑛 1 2 𝜅 𝑘 Pochhammer 𝑛 1 𝑘 𝑘 superscript 𝑧 𝑘 𝑧 digamma 1 2 𝑛 1 2 𝜅 𝑘 digamma 1 𝑘 digamma 𝑛 1 𝑘 {\displaystyle{\displaystyle W_{\kappa,\pm\frac{1}{2}n}\left(z\right)=\frac{(-% 1)^{n}e^{-\frac{1}{2}z}z^{\frac{1}{2}n+\frac{1}{2}}}{n!\Gamma\left(\frac{1}{2}% -\frac{1}{2}n-\kappa\right)}\left(\sum_{k=1}^{n}\frac{n!(k-1)!}{(n-k)!{\left(% \kappa+\frac{1}{2}-\frac{1}{2}n\right)_{k}}}z^{-k}-\sum_{k=0}^{\infty}\frac{{% \left(\frac{1}{2}n+\frac{1}{2}-\kappa\right)_{k}}}{{\left(n+1\right)_{k}}k!}z^% {k}\left(\ln z+\psi\left(\tfrac{1}{2}n+\tfrac{1}{2}-\kappa+k\right)-\psi\left(% 1+k\right)-\psi\left(n+1+k\right)\right)\right),}}
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    DLMF id
    DLMF:13.14.E8
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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}\neq 0,1,2,\dots}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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