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DLMF:15.9.E2 - MaRDI portal

DLMF:15.9.E2 (Q5789)

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DLMF:15.9.E2
No description defined

    Statements

    DLMF defining formula
    C n ( λ ) ⁡ ( x ) = ( 2 ⁢ λ ) n n ! ⁢ F ⁡ ( - n , n + 2 ⁢ λ λ + 1 2 ; 1 - x 2 ) . ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 Pochhammer 2 𝜆 𝑛 𝑛 Gauss-hypergeometric-F 𝑛 𝑛 2 𝜆 𝜆 1 2 1 𝑥 2 {\displaystyle{\displaystyle C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2% \lambda\right)_{n}}}{n!}F\left({-n,n+2\lambda\atop\lambda+\frac{1}{2}};\frac{1% -x}{2}\right).}}
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    DLMF 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.m2aadec
    0 references
    Q10733
    DLMF defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aadec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    ultraspherical (or Gegenbauer) polynomial
    DLMF 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
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C15.S1.XMD1.m1adec
    0 references
    integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1adec
    0 references

    Identifiers

     
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