DLMF:14.3.E18 (Q4707)

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DLMF:14.3.E18
No description defined

    Statements

    Defining formula
    P ν - μ ⁡ ( x ) = 2 - μ ⁢ x ν - μ ⁢ ( x 2 - 1 ) μ / 2 ⁢ 𝐅 ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν , 1 2 ⁢ μ - 1 2 ⁢ ν + 1 2 ; μ + 1 ; 1 - 1 x 2 ) , Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 2 𝜇 superscript 𝑥 𝜈 𝜇 superscript superscript 𝑥 2 1 𝜇 2 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 1 2 𝜇 1 2 𝜈 1 2 𝜇 1 1 1 superscript 𝑥 2 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}x^{\nu-\mu}% \left(x^{2}-1\right)^{\mu/2}\mathbf{F}\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,% \tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};\mu+1;1-\frac{1}{x^{2}}\right),}}
    0 references
    DLMF id
    DLMF:14.3.E18
    0 references
    Symbols used
    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.m1afdec
    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.m2andec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1qdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1mdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1pdec
    0 references
     
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