DLMF:14.3.E17 (Q4706)

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DLMF:14.3.E17
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    Statements

    Defining formula
    P ν - μ ⁡ ( x ) = π ⁢ ( x 2 - 1 ) μ / 2 2 μ ⁢ ( 𝐅 ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν , 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ; 1 2 ; x 2 ) Γ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) - x ⁢ 𝐅 ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 , 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ; 3 2 ; x 2 ) Γ ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν ) ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ) ) , Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜋 superscript superscript 𝑥 2 1 𝜇 2 superscript 2 𝜇 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 1 2 𝜈 1 2 𝜇 1 2 1 2 superscript 𝑥 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 𝑥 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 1 2 1 2 𝜈 1 2 𝜇 1 3 2 superscript 𝑥 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=\frac{\pi\left(x^{2}% -1\right)^{\mu/2}}{2^{\mu}}\left(\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}% {2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{% \Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{% 1}{2}\nu+\frac{1}{2}\mu+1\right)}-\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{% 1}{2}\nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}% {\Gamma\left(\frac{1}{2}\mu-\frac{1}{2}\nu\right)\Gamma\left(\frac{1}{2}\nu+% \frac{1}{2}\mu+\frac{1}{2}\right)}\right),}}
    0 references
    DLMF id
    DLMF:14.3.E17
    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.m2ajdec
    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.m1aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2amdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1pdec
    0 references
     
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