DLMF:16.5.E3 (Q5214)

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DLMF:16.5.E3
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    Statements

    Defining formula
    F q p + 1 ⁡ ( a 0 , … , a p b 1 , … , b q ; z ) = 1 Γ ⁡ ( a 0 ) ⁢ ∫ 0 ∞ e - t ⁢ t a 0 - 1 ⁢ F q p ⁡ ( a 1 , … , a p b 1 , ⋯ , b q ; z ⁢ t ) ⁢ d t , Gauss-hypergeometric-pFq 𝑝 1 𝑞 subscript 𝑎 0 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 1 Euler-Gamma subscript 𝑎 0 superscript subscript 0 𝑡 superscript 𝑡 subscript 𝑎 0 1 Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 ⋯ subscript 𝑏 𝑞 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle{{}_{p+1}F_{q}}\left({a_{0},\dots,a_{p}\atop b_{1}% ,\dots,b_{q}};z\right)=\frac{1}{\Gamma\left(a_{0}\right)}\int_{0}^{\infty}{% \mathrm{e}^{-t}}t^{a_{0}-1}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},% \cdots,b_{q}};zt\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:16.5.E3
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    constraint
    ℜ ⁡ z < 1 𝑧 1 {\displaystyle{\displaystyle\Re z<1}}
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    ℜ ⁡ a 0 > 0 subscript 𝑎 0 0 {\displaystyle{\displaystyle\Re a_{0}>0}}
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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.m2abdec
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    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.m1abdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3abdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11690
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1bdec
    0 references
    Q11691
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1bdec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1bdec
    0 references
     
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