DLMF:18.5.E8 (Q5516)

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DLMF:18.5.E8
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    Statements

    Defining formula
    P n ( α , β ) ⁡ ( x ) = 2 - n ⁢ ∑ ℓ = 0 n ( n + α ℓ ) ⁢ ( n + β n - ℓ ) ⁢ ( x - 1 ) n - ℓ ⁢ ( x + 1 ) ℓ = ( α + 1 ) n n ! ⁢ ( x + 1 2 ) n ⁢ F 1 2 ⁡ ( - n , - n - β α + 1 ; x - 1 x + 1 ) , Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 2 𝑛 superscript subscript ℓ 0 𝑛 binomial 𝑛 𝛼 ℓ binomial 𝑛 𝛽 𝑛 ℓ superscript 𝑥 1 𝑛 ℓ superscript 𝑥 1 ℓ Pochhammer 𝛼 1 𝑛 𝑛 superscript 𝑥 1 2 𝑛 Gauss-hypergeometric-F-as-2F1 𝑛 𝑛 𝛽 𝛼 1 𝑥 1 𝑥 1 {\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{% \ell=0}^{n}\genfrac{(}{)}{0.0pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+% \beta}{n-\ell}(x-1)^{n-\ell}(x+1)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!% }\left(\frac{x+1}{2}\right)^{n}{{}_{2}F_{1}}\left({-n,-n-\beta\atop\alpha+1};% \frac{x-1}{x+1}\right),}}
    0 references
    DLMF id
    DLMF:18.5.E8
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5aadec
    0 references
    Q11660
    Defining formula
    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.m2abdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1abdec
    0 references
    Q10754
    Defining formula
    ( m n ) binomial 𝑚 𝑛 {\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
    xml-id
    C1.S2.SS1.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1adec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1hdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1hdec
    0 references
     
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