DLMF:18.12.E3 (Q5645)

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DLMF:18.12.E3
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    Statements

    Defining formula
    ( 1 + z ) - α - β - 1 ⁢ F 1 2 ⁡ ( 1 2 ⁢ ( α + β + 1 ) , 1 2 ⁢ ( α + β + 2 ) β + 1 ; 2 ⁢ ( x + 1 ) ⁢ z ( 1 + z ) 2 ) = ∑ n = 0 ∞ ( α + β + 1 ) n ( β + 1 ) n ⁢ P n ( α , β ) ⁡ ( x ) ⁢ z n , superscript 1 𝑧 𝛼 𝛽 1 Gauss-hypergeometric-F-as-2F1 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝛽 1 2 𝑥 1 𝑧 superscript 1 𝑧 2 superscript subscript 𝑛 0 Pochhammer 𝛼 𝛽 1 𝑛 Pochhammer 𝛽 1 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑧 𝑛 {\displaystyle{\displaystyle(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({% \tfrac{1}{2}(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2% (x+1)z}{(1+z)^{2}}\right)=\sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right% )_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},}}
    0 references
    DLMF id
    DLMF:18.12.E3
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    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
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    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.m5adec
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    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.m1adec
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    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1bdec
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    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1bdec
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    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1bdec
    0 references
     
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