DLMF:18.12.E4 (Q5646)

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DLMF:18.12.E4
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    Statements

    Defining formula
    ( 1 - 2 ⁢ x ⁢ z + z 2 ) - λ = ∑ n = 0 ∞ C n ( λ ) ⁡ ( x ) ⁢ z n = ∑ n = 0 ∞ ( 2 ⁢ λ ) n ( λ + 1 2 ) n ⁢ P n ( λ - 1 2 , λ - 1 2 ) ⁡ ( x ) ⁢ z n , superscript 1 2 𝑥 𝑧 superscript 𝑧 2 𝜆 superscript subscript 𝑛 0 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 𝑧 𝑛 superscript subscript 𝑛 0 Pochhammer 2 𝜆 𝑛 Pochhammer 𝜆 1 2 𝑛 Jacobi-polynomial-P 𝜆 1 2 𝜆 1 2 𝑛 𝑥 superscript 𝑧 𝑛 {\displaystyle{\displaystyle(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(% \lambda)}_{n}\left(x\right)z^{n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda% \right)_{n}}}{{\left(\lambda+\tfrac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},% \lambda-\frac{1}{2})}_{n}\left(x\right)z^{n},}}
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    DLMF id
    DLMF:18.12.E4
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    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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    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
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    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
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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.m2adec
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    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1cdec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1cdec
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