DLMF:16.3.E2 (Q5189)

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DLMF:16.3.E2
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    Statements

    Defining formula
    d n d z n ⁡ ( z γ ⁢ F q p ⁡ ( a 1 , … , a p b 1 , … , b q ; z ) ) = ( γ - n + 1 ) n ⁢ z γ - n ⁢ F q + 1 p + 1 ⁡ ( γ + 1 , a 1 , … , a p γ + 1 - n , b 1 , … , b q ; z ) , derivative 𝑧 𝑛 superscript 𝑧 𝛾 Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 Pochhammer 𝛾 𝑛 1 𝑛 superscript 𝑧 𝛾 𝑛 Gauss-hypergeometric-pFq 𝑝 1 𝑞 1 𝛾 1 subscript 𝑎 1 … subscript 𝑎 𝑝 𝛾 1 𝑛 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {\gamma}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)% \right)={\left(\gamma-n+1\right)_{n}}z^{\gamma-n}{{}_{p+1}F_{q+1}}\left({% \gamma+1,a_{1},\dots,a_{p}\atop\gamma+1-n,b_{1},\dots,b_{q}};z\right),}}
    0 references
    DLMF id
    DLMF:16.3.E2
    0 references
    Symbols used
    Q11125
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1aadec
    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
    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.m2aadec
    0 references
    Q11690
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1adec
    0 references
    Q11691
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1adec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1adec
    0 references
    Q11693
    Defining formula
    a , a 1 , … , a p 𝑎 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}
    xml-id
    C16.S1.XMD4.m1adec
    0 references
     
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