DLMF:16.19.E6 (Q5286)

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DLMF:16.19.E6
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    Statements

    Defining formula
    ∫ 0 1 t - a 0 ⁢ ( 1 - t ) a 0 - b q + 1 - 1 ⁢ G p , q m , n ⁡ ( z ⁢ t ; a 1 , … , a p b 1 , … , b q ) ⁢ d t = Γ ⁡ ( a 0 - b q + 1 ) ⁢ G p + 1 , q + 1 m , n + 1 ⁡ ( z ; a 0 , … , a p b 1 , … , b q + 1 ) , superscript subscript 0 1 superscript 𝑡 subscript 𝑎 0 superscript 1 𝑡 subscript 𝑎 0 subscript 𝑏 𝑞 1 1 Meijer-G 𝑚 𝑛 𝑝 𝑞 𝑧 𝑡 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑡 Euler-Gamma subscript 𝑎 0 subscript 𝑏 𝑞 1 Meijer-G 𝑚 𝑛 1 𝑝 1 𝑞 1 𝑧 subscript 𝑎 0 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 1 {\displaystyle{\displaystyle\int_{0}^{1}t^{-a_{0}}(1-t)^{a_{0}-b_{q+1}-1}{G^{m% ,n}_{p,q}}\left(zt;{a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}}\right)\mathrm{d}% t=\Gamma\left(a_{0}-b_{q+1}\right){G^{m,n+1}_{p+1,q+1}}\left(z;{a_{0},\dots,a_% {p}\atop b_{1},\dots,b_{q+1}}\right),}}
    0 references
    DLMF id
    DLMF:16.19.E6
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Meijer $$G$$ -function
    Defining formula
    G p , q m , n ⁡ ( z ; a 1 , … , a p b 1 , … , b q ) Meijer-G 𝑚 𝑛 𝑝 𝑞 𝑧 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle{G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(% \NVar{z};{\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)}}
    xml-id
    C16.S17.E1.m2aedec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
     
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