DLMF:15.8.E29 (Q5086)

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DLMF:15.8.E29
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    Statements

    Defining formula
    F ⁡ ( a , 1 3 ⁢ a + 1 3 2 3 ⁢ a + 2 3 ; z ) = ( 1 + z ) - 2 ⁢ a ⁢ F ⁡ ( a , 2 3 ⁢ a + 1 6 4 3 ⁢ a + 1 3 ; 4 ⁢ z ( 1 + z ) 2 ) . Gauss-hypergeometric-F 𝑎 1 3 𝑎 1 3 2 3 𝑎 2 3 𝑧 superscript 1 𝑧 2 𝑎 Gauss-hypergeometric-F 𝑎 2 3 𝑎 1 6 4 3 𝑎 1 3 4 𝑧 superscript 1 𝑧 2 {\displaystyle{\displaystyle F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2% }{3}a+\tfrac{2}{3}};z\right)=\left(1+\sqrt{z}\right)^{-2a}\*F\left({a,\tfrac{2% }{3}a+\tfrac{1}{6}\atop\tfrac{4}{3}a+\tfrac{1}{3}};\frac{4\sqrt{z}}{(1+\sqrt{z% })^{2}}\right).}}
    0 references
    DLMF id
    DLMF:15.8.E29
    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.m2ardec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1abdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1aadec
    0 references
     
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