DLMF:14.3.E22 (Q4711)

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DLMF:14.3.E22
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    Statements

    Defining formula
    P ν μ ⁡ ( x ) = 2 μ ⁢ Γ ⁡ ( 1 - 2 ⁢ μ ) ⁢ Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 1 ) ⁢ Γ ⁡ ( 1 - μ ) ⁢ ( x 2 - 1 ) μ / 2 ⁢ C ν + μ ( 1 2 - μ ) ⁡ ( x ) . Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 2 𝜇 Euler-Gamma 1 2 𝜇 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 1 𝜇 superscript superscript 𝑥 2 1 𝜇 2 ultraspherical-Gegenbauer-polynomial 1 2 𝜇 𝜈 𝜇 𝑥 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma% \left(1-2\mu\right)\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)% \Gamma\left(1-\mu\right)\left(x^{2}-1\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{% \nu+\mu}\left(x\right).}}
    0 references
    DLMF id
    DLMF:14.3.E22
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2andec
    0 references
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1agdec
    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.m2aadec
    0 references
     
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