DLMF:15.8.E32 (Q5089)

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DLMF:15.8.E32
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    Statements

    test
    ( 1 - z 3 ) a ( - z ) 3 ⁢ a ⁢ ( 1 Γ ⁡ ( a + 2 3 ) ⁢ Γ ⁡ ( 2 3 ) ⁢ F ⁡ ( a , a + 1 3 2 3 ; z - 3 ) + e 1 3 ⁢ π ⁢ i z ⁢ Γ ⁡ ( a ) ⁢ Γ ⁡ ( 4 3 ) ⁢ F ⁡ ( a + 1 3 , a + 2 3 4 3 ; z - 3 ) ) = 3 3 2 ⁢ a + 1 2 ⁢ e 1 2 ⁢ a ⁢ π ⁢ i ⁢ Γ ⁡ ( a + 1 3 ) ⁢ ( 1 - ζ ) a 2 ⁢ π ⁢ Γ ⁡ ( 2 ⁢ a + 2 3 ) ⁢ ( - ζ ) 2 ⁢ a ⁢ F ⁡ ( a + 1 3 , 3 ⁢ a 2 ⁢ a + 2 3 ; ζ - 1 ) , superscript 1 superscript 𝑧 3 𝑎 superscript 𝑧 3 𝑎 1 Euler-Gamma 𝑎 2 3 Euler-Gamma 2 3 Gauss-hypergeometric-F 𝑎 𝑎 1 3 2 3 superscript 𝑧 3 superscript 𝑒 1 3 𝜋 imaginary-unit 𝑧 Euler-Gamma 𝑎 Euler-Gamma 4 3 Gauss-hypergeometric-F 𝑎 1 3 𝑎 2 3 4 3 superscript 𝑧 3 superscript 3 3 2 𝑎 1 2 superscript 𝑒 1 2 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑎 1 3 superscript 1 𝜁 𝑎 2 𝜋 Euler-Gamma 2 𝑎 2 3 superscript 𝜁 2 𝑎 Gauss-hypergeometric-F 𝑎 1 3 3 𝑎 2 𝑎 2 3 superscript 𝜁 1 {\displaystyle{\displaystyle\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3% a}}\left(\frac{1}{\Gamma\left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}% \right)}F\left({a,a+\frac{1}{3}\atop\frac{2}{3}};z^{-3}\right)+\frac{e^{\frac{% 1}{3}\pi\mathrm{i}}}{z\Gamma\left(a\right)\Gamma\left(\frac{4}{3}\right)}F% \left({a+\frac{1}{3},a+\frac{2}{3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac% {3^{\frac{3}{2}a+\frac{1}{2}}e^{\frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{% 1}{3}\right)(1-\zeta)^{a}}{2\pi\Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}% F\left({a+\frac{1}{3},3a\atop 2a+\frac{2}{3}};\zeta^{-1}\right),}}
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    DLMF id
    DLMF:15.8.E32
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    constraint
    | ph ⁡ ( - z ) | < 1 3 ⁢ π phase 𝑧 1 3 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi}}
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    | z | > 1 𝑧 1 {\displaystyle{\displaystyle|z|>1}}
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    | ph ⁡ ( - z ) | < 1 3 ⁢ π ph 𝑧 1 3 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2amdec
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    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    test
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2audec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2asdec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1asdec
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    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1aedec
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    Q11645
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1addec
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    Q11659
    test
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C15.S8.XMD1.m1dec
    0 references
     
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