DLMF:16.5.E1 (Q5212): Difference between revisions

From DRMF
Jump to navigation Jump to search
← Older editNewer edit →
‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11125 / rank
 
Normal rank
Property / Symbols used: Q11125 / qualifier
 
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)}}

\genhyperF{\NVar{p}}{\NVar{q}}@{\NVar{a_{1},\dots,a_{p}}}{\NVar{b_{1},\dots,b_{q}}}{\NVar{z}}
Property / Symbols used: Q11125 / qualifier
 
xml-id: C16.S2.m1adec

Revision as of 14:33, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:16.5.E1
No description defined

    Statements

    Defining formula
    ( ∏ k = 1 p Γ ⁡ ( a k ) / ∏ k = 1 q Γ ⁡ ( b k ) ) ⁢ F q p ⁡ ( a 1 , … , a p b 1 , … , b q ; z ) = 1 2 ⁢ π ⁢ i ⁢ ∫ L ( ∏ k = 1 p Γ ⁡ ( a k + s ) / ∏ k = 1 q Γ ⁡ ( b k + s ) ) ⁢ Γ ⁡ ( - s ) ⁢ ( - z ) s ⁢ d s , superscript subscript product 𝑘 1 𝑝 Euler-Gamma subscript 𝑎 𝑘 superscript subscript product 𝑘 1 𝑞 Euler-Gamma subscript 𝑏 𝑘 Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 1 2 𝜋 imaginary-unit subscript 𝐿 superscript subscript product 𝑘 1 𝑝 Euler-Gamma subscript 𝑎 𝑘 𝑠 superscript subscript product 𝑘 1 𝑞 Euler-Gamma subscript 𝑏 𝑘 𝑠 Euler-Gamma 𝑠 superscript 𝑧 𝑠 𝑠 {\displaystyle{\displaystyle\left({\textstyle\ifrac{\prod\limits_{k=1}^{p}% \Gamma\left(a_{k}\right)}{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}% \right){{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=% \frac{1}{2\pi\mathrm{i}}\int_{L}\left({\textstyle\ifrac{\prod\limits_{k=1}^{p}% \Gamma\left(a_{k}+s\right)}{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}+s\right)}}% \right)\Gamma\left(-s\right)(-z)^{s}\mathrm{d}s,}}
    0 references
    DLMF id
    DLMF:16.5.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
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5212&oldid=48312"