DLMF:15.9.E3 (Q5093)

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DLMF:15.9.E3
No description defined

    Statements

    Defining formula
    C n ( λ ) ⁡ ( x ) = ( 2 ⁢ x ) n ⁢ ( λ ) n n ! ⁢ F ⁡ ( - 1 2 ⁢ n , 1 2 ⁢ ( 1 - n ) 1 - λ - n ; 1 x 2 ) . ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 superscript 2 𝑥 𝑛 Pochhammer 𝜆 𝑛 𝑛 Gauss-hypergeometric-F 1 2 𝑛 1 2 1 𝑛 1 𝜆 𝑛 1 superscript 𝑥 2 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=(2x)^{n}\frac{{% \left(\lambda\right)_{n}}}{n!}F\left({-\frac{1}{2}n,\frac{1}{2}(1-n)\atop 1-% \lambda-n};\frac{1}{x^{2}}\right).}}
    0 references
    DLMF id
    DLMF:15.9.E3
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    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.m2abdec
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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.m1abdec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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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.m2aadec
    0 references
    Q11649
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C15.S1.XMD1.m1bdec
    0 references
    Q11650
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1bdec
    0 references
     
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