DLMF:18.18.E14 (Q5804)

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DLMF:18.18.E14
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    Statements

    Defining formula
    P n ( γ , β ) ⁡ ( x ) = ( β + 1 ) n ( α + β + 2 ) n ⁢ ∑ ℓ = 0 n α + β + 2 ⁢ ℓ + 1 α + β + 1 ⁢ ( α + β + 1 ) ℓ ⁢ ( n + β + γ + 1 ) ℓ ( β + 1 ) ℓ ⁢ ( n + α + β + 2 ) ℓ ⁢ ( γ - α ) n - ℓ ( n - ℓ ) ! ⁢ P ℓ ( α , β ) ⁡ ( x ) , Jacobi-polynomial-P 𝛾 𝛽 𝑛 𝑥 Pochhammer 𝛽 1 𝑛 Pochhammer 𝛼 𝛽 2 𝑛 superscript subscript ℓ 0 𝑛 𝛼 𝛽 2 ℓ 1 𝛼 𝛽 1 Pochhammer 𝛼 𝛽 1 ℓ Pochhammer 𝑛 𝛽 𝛾 1 ℓ Pochhammer 𝛽 1 ℓ Pochhammer 𝑛 𝛼 𝛽 2 ℓ Pochhammer 𝛾 𝛼 𝑛 ℓ 𝑛 ℓ Jacobi-polynomial-P 𝛼 𝛽 ℓ 𝑥 {\displaystyle{\displaystyle P^{(\gamma,\beta)}_{n}\left(x\right)=\dfrac{{% \left(\beta+1\right)_{n}}}{{\left(\alpha+\beta+2\right)_{n}}}\sum_{\ell=0}^{n}% \dfrac{\alpha+\beta+2\ell+1}{\alpha+\beta+1}\dfrac{{\left(\alpha+\beta+1\right% )_{\ell}}{\left(n+\beta+\gamma+1\right)_{\ell}}}{{\left(\beta+1\right)_{\ell}}% {\left(n+\alpha+\beta+2\right)_{\ell}}}\dfrac{{\left(\gamma-\alpha\right)_{n-% \ell}}}{(n-\ell)!}P^{(\alpha,\beta)}_{\ell}\left(x\right),}}
    0 references
    DLMF id
    DLMF:18.18.E14
    0 references
    Symbols used
    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.m2acdec
    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
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5agdec
    0 references
    Q11744
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C18.S1.XMD4.m1ddec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1mdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1jdec
    0 references
     
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