DLMF:15.8.E30 (Q5087)

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DLMF:15.8.E30
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    Statements

    Defining formula
    ( 1 - 1 2 ⁢ z ) - a ⁢ F ⁡ ( 1 2 ⁢ a , 1 2 ⁢ a + 1 2 1 3 ⁢ a + 5 6 ; ( z 2 - z ) 2 ) = F ⁡ ( a , 1 3 ⁢ a + 1 3 2 3 ⁢ a + 2 3 ; z ) = ( 1 + z ) - a ⁢ F ⁡ ( 1 2 ⁢ a , 1 2 ⁢ a + 1 2 2 3 ⁢ a + 2 3 ; 4 ⁢ z ( 1 + z ) 2 ) , superscript 1 1 2 𝑧 𝑎 Gauss-hypergeometric-F 1 2 𝑎 1 2 𝑎 1 2 1 3 𝑎 5 6 superscript 𝑧 2 𝑧 2 Gauss-hypergeometric-F 𝑎 1 3 𝑎 1 3 2 3 𝑎 2 3 𝑧 superscript 1 𝑧 𝑎 Gauss-hypergeometric-F 1 2 𝑎 1 2 𝑎 1 2 2 3 𝑎 2 3 4 𝑧 superscript 1 𝑧 2 {\displaystyle{\displaystyle\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}% {2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop\tfrac{1}{3}a+\tfrac{5}{6}};\left(\frac{z}% {2-z}\right)^{2}\right)=F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a% +\tfrac{2}{3}};z\right)=(1+z)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1% }{2}\atop\tfrac{2}{3}a+\tfrac{2}{3}};\frac{4z}{(1+z)^{2}}\right),}}
    0 references
    DLMF id
    DLMF:15.8.E30
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    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.m2asdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1acdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1abdec
    0 references
     
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