DLMF:16.4.E5 (Q5199): Difference between revisions

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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.m1abdec

Revision as of 14:30, 2 January 2020

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DLMF:16.4.E5
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    Statements

    Defining formula
    F 2 3 ⁡ ( - n , b , c 1 - b - n , 1 - c - n ; 1 ) = { 0 , n = 2 ⁢ k + 1 , ( 2 ⁢ k ) ! ⁢ Γ ⁡ ( b + k ) ⁢ Γ ⁡ ( c + k ) ⁢ Γ ⁡ ( b + c + 2 ⁢ k ) k ! ⁢ Γ ⁡ ( b + 2 ⁢ k ) ⁢ Γ ⁡ ( c + 2 ⁢ k ) ⁢ Γ ⁡ ( b + c + k ) , n = 2 ⁢ k , Gauss-hypergeometric-pFq 3 2 𝑛 𝑏 𝑐 1 𝑏 𝑛 1 𝑐 𝑛 1 cases 0 𝑛 2 𝑘 1 2 𝑘 Euler-Gamma 𝑏 𝑘 Euler-Gamma 𝑐 𝑘 Euler-Gamma 𝑏 𝑐 2 𝑘 𝑘 Euler-Gamma 𝑏 2 𝑘 Euler-Gamma 𝑐 2 𝑘 Euler-Gamma 𝑏 𝑐 𝑘 𝑛 2 𝑘 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({-n,b,c\atop 1-b-n,1-c-n};1% \right)=\begin{cases}0,&n=2k+1,\\ \dfrac{(2k)!\Gamma\left(b+k\right)\Gamma\left(c+k\right)\Gamma\left(b+c+2k% \right)}{k!\Gamma\left(b+2k\right)\Gamma\left(c+2k\right)\Gamma\left(b+c+k% \right)},&n=2k,\\ \end{cases}}}
    0 references
    DLMF id
    DLMF:16.4.E5
    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.m2aadec
    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
     
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