DLMF:15.8.E32 (Q5089)

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DLMF:15.8.E32
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    Statements

    Defining formula
    ( 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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