DLMF:16.18.E1 (Q5280): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11693 / rank
 
Normal rank
Property / Symbols used: Q11693 / qualifier
 
Defining formula:

a , a 1 , … , a p 𝑎 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}

a,a_{1},\ldots,a_{p}
Property / Symbols used: Q11693 / qualifier
 
xml-id: C16.S1.XMD4.m1dec

Revision as of 14:43, 2 January 2020

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DLMF:16.18.E1
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    Statements

    Defining formula
    F q p ⁡ ( a 1 , … , a p b 1 , … , b q ; z ) = ( ∏ k = 1 q Γ ⁡ ( b k ) / ∏ k = 1 p Γ ⁡ ( a k ) ) ⁢ G p , q + 1 1 , p ⁡ ( - z ; 1 - a 1 , … , 1 - a p 0 , 1 - b 1 , … , 1 - b q ) = ( ∏ k = 1 q Γ ⁡ ( b k ) / ∏ k = 1 p Γ ⁡ ( a k ) ) ⁢ G q + 1 , p p , 1 ⁡ ( - 1 z ; 1 , b 1 , … , b q a 1 , … , a p ) . Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 superscript subscript product 𝑘 1 𝑞 Euler-Gamma subscript 𝑏 𝑘 superscript subscript product 𝑘 1 𝑝 Euler-Gamma subscript 𝑎 𝑘 Meijer-G 1 𝑝 𝑝 𝑞 1 𝑧 1 subscript 𝑎 1 … 1 subscript 𝑎 𝑝 0 1 subscript 𝑏 1 … 1 subscript 𝑏 𝑞 superscript subscript product 𝑘 1 𝑞 Euler-Gamma subscript 𝑏 𝑘 superscript subscript product 𝑘 1 𝑝 Euler-Gamma subscript 𝑎 𝑘 Meijer-G 𝑝 1 𝑞 1 𝑝 1 𝑧 1 subscript 𝑏 1 … subscript 𝑏 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)=\left({\textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma% \left(b_{k}\right)}{\prod\limits_{k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^% {1,p}_{p,q+1}}\left(-z;{1-a_{1},\dots,1-a_{p}\atop 0,1-b_{1},\dots,1-b_{q}}% \right)=\left({\textstyle\ifrac{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)% }{\prod\limits_{k=1}^{p}\Gamma\left(a_{k}\right)}}\right){G^{p,1}_{q+1,p}}% \left(-\frac{1}{z};{1,b_{1},\dots,b_{q}\atop a_{1},\dots,a_{p}}\right).}}
    0 references
    DLMF id
    DLMF:16.18.E1
    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
    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.m1adec
    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.m2adec
    0 references
    Q11690
    Defining formula
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1dec
    0 references
    Q11691
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1dec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1dec
    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.m1dec
    0 references
     
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