DLMF:15.9.E15 (Q5105)

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DLMF:15.9.E15
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    Statements

    Defining formula
    C α ( λ ) ⁡ ( z ) = Γ ⁡ ( α + 2 ⁢ λ ) Γ ⁡ ( 2 ⁢ λ ) ⁢ Γ ⁡ ( α + 1 ) ⁢ F ⁡ ( - α , α + 2 ⁢ λ λ + 1 2 ; 1 - z 2 ) . ultraspherical-Gegenbauer-polynomial 𝜆 𝛼 𝑧 Euler-Gamma 𝛼 2 𝜆 Euler-Gamma 2 𝜆 Euler-Gamma 𝛼 1 Gauss-hypergeometric-F 𝛼 𝛼 2 𝜆 𝜆 1 2 1 𝑧 2 {\displaystyle{\displaystyle C^{(\lambda)}_{\alpha}\left(z\right)=\frac{\Gamma% \left(\alpha+2\lambda\right)}{\Gamma\left(2\lambda\right)\Gamma\left(\alpha+1% \right)}F\left({-\alpha,\alpha+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-z}{2% }\right).}}
    0 references
    DLMF id
    DLMF:15.9.E15
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2aldec
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    Q11475
    Defining formula
    C n ( λ ) ⁡ ( x ) ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 {\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r3.m2acdec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1dec
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