DLMF:13.15.E17 (Q4542)

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DLMF:13.15.E17
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    Statements

    Defining formula
    ( z ⁢ d d z ⁡ z ) n ⁢ ( e 1 2 ⁢ z ⁢ z - κ - 1 ⁢ M κ , μ ⁡ ( z ) ) = ( 1 2 + μ - κ ) n ⁢ e 1 2 ⁢ z ⁢ z n - κ - 1 ⁢ M κ - n , μ ⁡ ( z ) , superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝜅 1 Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 Pochhammer 1 2 𝜇 𝜅 𝑛 superscript 𝑒 1 2 𝑧 superscript 𝑧 𝑛 𝜅 1 Whittaker-confluent-hypergeometric-M 𝜅 𝑛 𝜇 𝑧 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(e^{\frac{1}{2}z}z^{-\kappa-1}M_{\kappa,\mu}\left(z\right)\right)={\left(% \tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-\kappa-1}M_{\kappa-n,% \mu}\left(z\right),}}
    0 references
    DLMF id
    DLMF:13.15.E17
    0 references
    Symbols used
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    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.m2aidec
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    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2abdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q11559
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1bdec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1pdec
    0 references
     
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