DLMF:16.5.E2 (Q5213): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11125 / rank
 
Normal rank
Property / Symbols used: Q11125 / qualifier
 
test:

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.m1aadec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aadec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aadec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1adec
Property / Symbols used
 
Property / Symbols used: Q11690 / rank
 
Normal rank
Property / Symbols used: Q11690 / qualifier
 
test:

p 𝑝 {\displaystyle{\displaystyle p}}

p
Property / Symbols used: Q11690 / qualifier
 
xml-id: C16.S1.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11691 / rank
 
Normal rank
Property / Symbols used: Q11691 / qualifier
 
test:

q 𝑞 {\displaystyle{\displaystyle q}}

q
Property / Symbols used: Q11691 / qualifier
 
xml-id: C16.S1.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Q11692 / rank
 
Normal rank
Property / Symbols used: Q11692 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11692 / qualifier
 
xml-id: C16.S1.XMD3.m1adec
Property / Symbols used
 
Property / Symbols used: Q11693 / rank
 
Normal rank
Property / Symbols used: Q11693 / qualifier
 
test:

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.m1adec
Property / Symbols used
 
Property / Symbols used: Q11694 / rank
 
Normal rank
Property / Symbols used: Q11694 / qualifier
 
test:

b , b 1 , … , b q 𝑏 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}

b,b_{1},\ldots,b_{q}
Property / Symbols used: Q11694 / qualifier
 
xml-id: C16.S1.XMD5.m1adec

Latest revision as of 13:33, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:16.5.E2
No description defined

    Statements

    test
    F q + 1 p + 1 ⁡ ( a 0 , … , a p b 0 , … , b q ; z ) = Γ ⁡ ( b 0 ) Γ ⁡ ( a 0 ) ⁢ Γ ⁡ ( b 0 - a 0 ) ⁢ ∫ 0 1 t a 0 - 1 ⁢ ( 1 - t ) b 0 - a 0 - 1 ⁢ F q p ⁡ ( a 1 , … , a p b 1 , … , b q ; z ⁢ t ) ⁢ d t , Gauss-hypergeometric-pFq 𝑝 1 𝑞 1 subscript 𝑎 0 … subscript 𝑎 𝑝 subscript 𝑏 0 … subscript 𝑏 𝑞 𝑧 Euler-Gamma subscript 𝑏 0 Euler-Gamma subscript 𝑎 0 Euler-Gamma subscript 𝑏 0 subscript 𝑎 0 superscript subscript 0 1 superscript 𝑡 subscript 𝑎 0 1 superscript 1 𝑡 subscript 𝑏 0 subscript 𝑎 0 1 Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle{{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{% 0},\dots,b_{q}};z\right)=\frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}% \right)\Gamma\left(b_{0}-a_{0}\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0% }-1}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};zt\right)% \mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:16.5.E2
    0 references
    constraint
    ℜ ⁡ b 0 > ℜ ⁡ a 0 > 0 subscript 𝑏 0 subscript 𝑎 0 0 {\displaystyle{\displaystyle\Re b_{0}>\Re a_{0}>0}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    Q11125
    test
    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.m1aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11690
    test
    p 𝑝 {\displaystyle{\displaystyle p}}
    xml-id
    C16.S1.XMD1.m1adec
    0 references
    Q11691
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1adec
    0 references
    Q11692
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1adec
    0 references
    Q11693
    test
    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.m1adec
    0 references
    Q11694
    test
    b , b 1 , … , b q 𝑏 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}
    xml-id
    C16.S1.XMD5.m1adec
    0 references
     
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