DLMF:14.3.E15 (Q4704)

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DLMF:14.3.E15
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    Statements

    Defining formula
    P ν - μ ⁡ ( x ) = 2 - μ ⁢ ( x 2 - 1 ) μ / 2 ⁢ 𝐅 ⁡ ( μ - ν , ν + μ + 1 ; μ + 1 ; 1 2 - 1 2 ⁢ x ) , Legendre-P-first-kind 𝜇 𝜈 𝑥 superscript 2 𝜇 superscript superscript 𝑥 2 1 𝜇 2 scaled-hypergeometric-bold-F 𝜇 𝜈 𝜈 𝜇 1 𝜇 1 1 2 1 2 𝑥 {\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-% 1\right)^{\mu/2}\mathbf{F}\left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}% {2}x\right),}}
    0 references
    DLMF id
    DLMF:14.3.E15
    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.m1acdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2akdec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1ndec
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    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1jdec
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    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1mdec
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