DLMF:14.3.E14 (Q4703)

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DLMF:14.3.E14
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    Statements

    Defining formula
    w 2 ⁢ ( ν , μ , x ) = 2 μ ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ + 1 2 ) ⁢ x ⁢ ( 1 - x 2 ) - μ / 2 ⁢ 𝐅 ⁡ ( 1 2 - 1 2 ⁢ ν - 1 2 ⁢ μ , 1 2 ⁢ ν - 1 2 ⁢ μ + 1 ; 3 2 ; x 2 ) . subscript 𝑤 2 𝜈 𝜇 𝑥 superscript 2 𝜇 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 2 𝑥 superscript 1 superscript 𝑥 2 𝜇 2 scaled-hypergeometric-bold-F 1 2 1 2 𝜈 1 2 𝜇 1 2 𝜈 1 2 𝜇 1 3 2 superscript 𝑥 2 {\displaystyle{\displaystyle w_{2}(\nu,\mu,x)=\frac{2^{\mu}\Gamma\left(\frac{1% }{2}\nu+\frac{1}{2}\mu+1\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+% \frac{1}{2}\right)}x\left(1-x^{2}\right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-% \tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2}% ;x^{2}\right).}}
    0 references
    DLMF id
    DLMF:14.3.E14
    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.m2ahdec
    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.m2ajdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1mdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1idec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1ldec
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    Q11594
    Defining formula
    w 2 subscript 𝑤 2 {\displaystyle{\displaystyle w_{2}}}
    xml-id
    C14.S3.XMD1.m1cdec
    0 references
     
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