DLMF:18.9.E20 (Q5616)

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DLMF:18.9.E20
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    Statements

    Defining formula
    d d x ⁡ ( ( 1 - x 2 ) λ - 1 2 ⁢ C n ( λ ) ⁡ ( x ) ) = - ( n + 1 ) ⁢ ( n + 2 ⁢ λ - 1 ) 2 ⁢ ( λ - 1 ) ⁢ ( 1 - x 2 ) λ - 3 2 ⁢ C n + 1 ( λ - 1 ) ⁡ ( x ) . derivative 𝑥 superscript 1 superscript 𝑥 2 𝜆 1 2 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑥 𝑛 1 𝑛 2 𝜆 1 2 𝜆 1 superscript 1 superscript 𝑥 2 𝜆 3 2 ultraspherical-Gegenbauer-polynomial 𝜆 1 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{% \lambda-\frac{1}{2}}C^{(\lambda)}_{n}\left(x\right)\right)=-\frac{(n+1)(n+2% \lambda-1)}{2(\lambda-1)}{(1-x^{2})^{\lambda-\frac{3}{2}}}C^{(\lambda-1)}_{n+1% }\left(x\right).}}
    0 references
    DLMF id
    DLMF:18.9.E20
    0 references
    Symbols used
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2aedec
    0 references
    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.m2addec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1tdec
    0 references
    Q11727
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1sdec
    0 references
     
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