DLMF:16.8.E8 (Q5227): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
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.m1cdec

Revision as of 14:36, 2 January 2020

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DLMF:16.8.E8
No description defined

    Statements

    Defining formula
    F q q + 1 ⁡ ( a 1 , … , a q + 1 b 1 , … , b q ; z ) = ∑ j = 1 q + 1 ( ∏ k = 1 k ≠ j q + 1 Γ ⁡ ( a k - a j ) Γ ⁡ ( a k ) / ∏ k = 1 q Γ ⁡ ( b k - a j ) Γ ⁡ ( b k ) ) ⁢ w ~ j ⁢ ( z ) , Gauss-hypergeometric-pFq 𝑞 1 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑞 1 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 superscript subscript 𝑗 1 𝑞 1 superscript subscript product 𝑘 1 𝑘 𝑗 𝑞 1 Euler-Gamma subscript 𝑎 𝑘 subscript 𝑎 𝑗 Euler-Gamma subscript 𝑎 𝑘 superscript subscript product 𝑘 1 𝑞 Euler-Gamma subscript 𝑏 𝑘 subscript 𝑎 𝑗 Euler-Gamma subscript 𝑏 𝑘 subscript ~ 𝑤 𝑗 𝑧 {\displaystyle{\displaystyle{{}_{q+1}F_{q}}\left({a_{1},\dots,a_{q+1}\atop b_{% 1},\dots,b_{q}};z\right)=\sum_{j=1}^{q+1}\left({\textstyle\ifrac{\prod\limits_% {\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\frac{\Gamma\left(a_{k}-a_{j}\right)}{\Gamma\left(% a_{k}\right)}}{\prod\limits_{k=1}^{q}\frac{\Gamma\left(b_{k}-a_{j}\right)}{% \Gamma\left(b_{k}\right)}}}\right)\widetilde{w}_{j}(z),}}
    0 references
    DLMF id
    DLMF:16.8.E8
    0 references
    constraint
    | ph ⁡ ( - z ) | ≤ π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|\leq\pi}}
    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.m1abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    0 references
    Q11691
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1edec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1gdec
    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.m1cdec
    0 references
     
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