DLMF:16.4.E17 (Q5211): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: bilateral hypergeometric function / rank
 
Normal rank
Property / Symbols used: bilateral hypergeometric function / qualifier
 
test:

H q p ⁡ ( a 1 , … , a p b 1 , … , b q ; z ) hypergeometric-pHq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}H_{\NVar{q}}}\left({\NVar{a_{1},% \dots,a_{p}}\atop\NVar{b_{1},\dots,b_{q}}};\NVar{z}\right)}}

\genhyperH{\NVar{p}}{\NVar{q}}@@{\NVar{a_{1},\dots,a_{p}}}{\NVar{b_{1},\dots,b_{q}}}{\NVar{z}}
Property / Symbols used: bilateral hypergeometric function / qualifier
 
xml-id: C16.S4.E16.m2aadec

Revision as of 13:32, 2 January 2020

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DLMF:16.4.E17
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    Statements

    test
    H 2 2 ⁡ ( a , b c , d ; 1 ) = Γ ⁡ ( c ) ⁢ Γ ⁡ ( d ) ⁢ Γ ⁡ ( 1 - a ) ⁢ Γ ⁡ ( 1 - b ) ⁢ Γ ⁡ ( c + d - a - b - 1 ) Γ ⁡ ( c - a ) ⁢ Γ ⁡ ( d - a ) ⁢ Γ ⁡ ( c - b ) ⁢ Γ ⁡ ( d - b ) , hypergeometric-pHq 2 2 𝑎 𝑏 𝑐 𝑑 1 Euler-Gamma 𝑐 Euler-Gamma 𝑑 Euler-Gamma 1 𝑎 Euler-Gamma 1 𝑏 Euler-Gamma 𝑐 𝑑 𝑎 𝑏 1 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑑 𝑎 Euler-Gamma 𝑐 𝑏 Euler-Gamma 𝑑 𝑏 {\displaystyle{\displaystyle{{}_{2}H_{2}}\left({a,b\atop c,d};1\right)=\frac{% \Gamma\left(c\right)\Gamma\left(d\right)\Gamma\left(1-a\right)\Gamma\left(1-b% \right)\Gamma\left(c+d-a-b-1\right)}{\Gamma\left(c-a\right)\Gamma\left(d-a% \right)\Gamma\left(c-b\right)\Gamma\left(d-b\right)},}}
    0 references
    DLMF id
    DLMF:16.4.E17
    0 references
    constraint
    ℜ ⁡ ( c + d - a - b ) > 1 𝑐 𝑑 𝑎 𝑏 1 {\displaystyle{\displaystyle\Re(c+d-a-b)>1}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
    0 references
    bilateral hypergeometric function
    test
    H q p ⁡ ( a 1 , … , a p b 1 , … , b q ; z ) hypergeometric-pHq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}H_{\NVar{q}}}\left({\NVar{a_{1},% \dots,a_{p}}\atop\NVar{b_{1},\dots,b_{q}}};\NVar{z}\right)}}
    xml-id
    C16.S4.E16.m2aadec
    0 references
     
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