DLMF:15.8.E24 (Q5081)

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DLMF:15.8.E24
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    Statements

    test
    F ⁡ ( a , b a - b + 1 ; z ) = ( 1 - z ) - a ⁢ Γ ⁡ ( a - b + 1 ) ⁢ Γ ⁡ ( 1 2 ) Γ ⁡ ( 1 2 ⁢ a + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ a - b + 1 ) ⁢ F ⁡ ( 1 2 ⁢ a , 1 2 ⁢ a - b + 1 2 1 2 ; ( z + 1 z - 1 ) 2 ) + ( 1 + z ) ⁢ ( 1 - z ) - a - 1 ⁢ Γ ⁡ ( a - b + 1 ) ⁢ Γ ⁡ ( - 1 2 ) Γ ⁡ ( 1 2 ⁢ a ) ⁢ Γ ⁡ ( 1 2 ⁢ a - b + 1 2 ) ⁢ F ⁡ ( 1 2 ⁢ a + 1 2 , 1 2 ⁢ a - b + 1 3 2 ; ( z + 1 z - 1 ) 2 ) , Gauss-hypergeometric-F 𝑎 𝑏 𝑎 𝑏 1 𝑧 superscript 1 𝑧 𝑎 Euler-Gamma 𝑎 𝑏 1 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑎 1 2 Euler-Gamma 1 2 𝑎 𝑏 1 Gauss-hypergeometric-F 1 2 𝑎 1 2 𝑎 𝑏 1 2 1 2 superscript 𝑧 1 𝑧 1 2 1 𝑧 superscript 1 𝑧 𝑎 1 Euler-Gamma 𝑎 𝑏 1 Euler-Gamma 1 2 Euler-Gamma 1 2 𝑎 Euler-Gamma 1 2 𝑎 𝑏 1 2 Gauss-hypergeometric-F 1 2 𝑎 1 2 1 2 𝑎 𝑏 1 3 2 superscript 𝑧 1 𝑧 1 2 {\displaystyle{\displaystyle F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}\frac{% \Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}% {2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}F\left({\tfrac{1}% {2}a,\tfrac{1}{2}a-b+\tfrac{1}{2}\atop\tfrac{1}{2}};\left(\frac{z+1}{z-1}% \right)^{2}\right)+(1+z)(1-z)^{-a-1}\frac{\Gamma\left(a-b+1\right)\Gamma\left(% -\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}% a-b+\tfrac{1}{2}\right)}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}a-b+1% \atop\tfrac{3}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right),}}
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    DLMF id
    DLMF:15.8.E24
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    constraint
    | ph ⁡ ( - z ) | < π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}\left(-z\right)|<\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.m2ahdec
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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.m2amdec
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