DLMF:13.3.E20 (Q4354)

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DLMF:13.3.E20
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    Statements

    Defining formula
    d n d z n ⁡ ( e - z ⁢ M ⁡ ( a , b , z ) ) = ( - 1 ) n ⁢ ( b - a ) n ( b ) n ⁢ e - z ⁢ M ⁡ ( a , b + n , z ) , derivative 𝑧 𝑛 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑧 superscript 1 𝑛 Pochhammer 𝑏 𝑎 𝑛 Pochhammer 𝑏 𝑛 superscript 𝑒 𝑧 Kummer-confluent-hypergeometric-M 𝑎 𝑏 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^% {-z}M\left(a,b,z\right)\right)=(-1)^{n}\frac{{\left(b-a\right)_{n}}}{{\left(b% \right)_{n}}}e^{-z}M\left(a,b+n,z\right),}}
    0 references
    DLMF id
    DLMF:13.3.E20
    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.m2aldec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1addec
    0 references
    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.m2aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q11559
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C13.S1.XMD2.m1ddec
    0 references
    Q11557
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C13.S1.XMD6.m1sdec
    0 references
     
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