DLMF:15.9.E1 (Q5091)

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DLMF:15.9.E1
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    Statements

    test
    P n ( α , β ) ⁡ ( x ) = ( α + 1 ) n n ! ⁢ F ⁡ ( - n , n + α + β + 1 α + 1 ; 1 - x 2 ) . Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Pochhammer 𝛼 1 𝑛 𝑛 Gauss-hypergeometric-F 𝑛 𝑛 𝛼 𝛽 1 𝛼 1 1 𝑥 2 {\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=\frac{{\left% (\alpha+1\right)_{n}}}{n!}F\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}% {2}\right).}}
    0 references
    DLMF id
    DLMF:15.9.E1
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    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2adec
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    Q11660
    test
    P n ( α , β ) ⁡ ( x ) Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(% \NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r2.m2adec
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    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
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    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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