DLMF:16.11.E6 (Q5240): Difference between revisions

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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.m1adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
test:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10817 / rank
 
Normal rank
Property / Symbols used: Q10817 / qualifier
 
test:

ph phase {\displaystyle{\displaystyle\operatorname{ph}}}

\phase
Property / Symbols used: Q10817 / qualifier
 
xml-id: C1.S9.E7.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.m1ddec
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.m1bdec
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.m1cdec
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.m1cdec
Property / Symbols used
 
Property / Symbols used: formal infinite series (locally) / rank
 
Normal rank
Property / Symbols used: formal infinite series (locally) / qualifier
 
test:

H p , q ⁢ ( z ) subscript 𝐻 𝑝 𝑞 𝑧 {\displaystyle{\displaystyle H_{p,q}(z)}}

H_{p,q}(z)
Property / Symbols used: formal infinite series (locally) / qualifier
 
xml-id: C16.S11.XMD2.m1adec

Latest revision as of 13:38, 2 January 2020

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DLMF:16.11.E6
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    Statements

    test
    ( ∏ ℓ = 1 q + 1 Γ ⁡ ( a ℓ ) / ∏ ℓ = 1 q Γ ⁡ ( b ℓ ) ) ⁢ F q q + 1 ⁡ ( a 1 , … , a q + 1 b 1 , … , b q ; z ) = H q + 1 , q ⁢ ( - z ) , superscript subscript product ℓ 1 𝑞 1 Euler-Gamma subscript 𝑎 ℓ superscript subscript product ℓ 1 𝑞 Euler-Gamma subscript 𝑏 ℓ Gauss-hypergeometric-pFq 𝑞 1 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑞 1 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 subscript 𝐻 𝑞 1 𝑞 𝑧 {\displaystyle{\displaystyle\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q+1% }\Gamma\left(a_{\ell}\right)}{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}% \right)}}\right){{}_{q+1}F_{q}}\left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{% q}};z\right)=H_{q+1,q}(-z),}}
    0 references
    DLMF id
    DLMF:16.11.E6
    0 references
    constraint
    | ph ⁡ ( - z ) | ≤ π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|\leq\pi}}
    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.m1adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1adec
    0 references
    Q11691
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1ddec
    0 references
    Q11692
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1bdec
    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.m1cdec
    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.m1cdec
    0 references
    formal infinite series (locally)
    test
    H p , q ⁢ ( z ) subscript 𝐻 𝑝 𝑞 𝑧 {\displaystyle{\displaystyle H_{p,q}(z)}}
    xml-id
    C16.S11.XMD2.m1adec
    0 references
     
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