DLMF:16.2.E5 (Q5187)

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DLMF:16.2.E5
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    Statements

    Defining formula
    𝐅 q p ⁑ ( 𝐚 ; 𝐛 ; z ) = F q p ⁑ ( a 1 , … , a p b 1 , … , b q ; z ) / ( Ξ“ ⁑ ( b 1 ) ⁒ β‹― ⁒ Ξ“ ⁑ ( b q ) ) = βˆ‘ k = 0 ∞ ( a 1 ) k ⁒ β‹― ⁒ ( a p ) k Ξ“ ⁑ ( b 1 + k ) ⁒ β‹― ⁒ Ξ“ ⁑ ( b q + k ) ⁒ z k k ! ; hypergeometric-bold-pFq 𝑝 π‘ž 𝐚 𝐛 𝑧 Gauss-hypergeometric-pFq 𝑝 π‘ž subscript π‘Ž 1 … subscript π‘Ž 𝑝 subscript 𝑏 1 … subscript 𝑏 π‘ž 𝑧 Euler-Gamma subscript 𝑏 1 β‹― Euler-Gamma subscript 𝑏 π‘ž superscript subscript π‘˜ 0 Pochhammer subscript π‘Ž 1 π‘˜ β‹― Pochhammer subscript π‘Ž 𝑝 π‘˜ Euler-Gamma subscript 𝑏 1 π‘˜ β‹― Euler-Gamma subscript 𝑏 π‘ž π‘˜ superscript 𝑧 π‘˜ π‘˜ {\displaystyle{\displaystyle{{}_{p}{\mathbf{F}}_{q}}\left(\mathbf{a};\mathbf{b% };z\right)=\ifrac{{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}% };z\right)}{\left(\Gamma\left(b_{1}\right)\cdots\Gamma\left(b_{q}\right)\right% )}=\sum_{k=0}^{\infty}\frac{{\left(a_{1}\right)_{k}}\cdots{\left(a_{p}\right)_% {k}}}{\Gamma\left(b_{1}+k\right)\cdots\Gamma\left(b_{q}+k\right)}\frac{z^{k}}{% k!};}}
    0 references
    DLMF id
    DLMF:16.2.E5
    0 references
    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q11125
    Defining formula
    F q p ⁑ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 π‘ž subscript π‘Ž 1 … subscript π‘Ž 𝑝 subscript 𝑏 1 … subscript 𝑏 π‘ž 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.m1acdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer π‘Ž 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    Q11690
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1cdec
    0 references
    Q11691
    Defining formula
    q π‘ž {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1ddec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1cdec
    0 references
    Q11693
    Defining formula
    a , a 1 , … , a p π‘Ž subscript π‘Ž 1 … subscript π‘Ž 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}
    xml-id
    C16.S1.XMD4.m1bdec
    0 references
     
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