DLMF:14.3.E4 (Q4693)

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DLMF:14.3.E4
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    Statements

    Defining formula
    𝖯 Ξ½ m ⁑ ( x ) = ( - 1 ) m ⁒ Ξ“ ⁑ ( Ξ½ + m + 1 ) 2 m ⁒ Ξ“ ⁑ ( Ξ½ - m + 1 ) ⁒ ( 1 - x 2 ) m / 2 ⁒ 𝐅 ⁑ ( Ξ½ + m + 1 , m - Ξ½ ; m + 1 ; 1 2 - 1 2 ⁒ x ) ; Ferrers-Legendre-P-first-kind π‘š 𝜈 π‘₯ superscript 1 π‘š Euler-Gamma 𝜈 π‘š 1 superscript 2 π‘š Euler-Gamma 𝜈 π‘š 1 superscript 1 superscript π‘₯ 2 π‘š 2 scaled-hypergeometric-bold-F 𝜈 π‘š 1 π‘š 𝜈 π‘š 1 1 2 1 2 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{% \Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(\nu-m+1\right)}\left(1-x^{2}% \right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x% \right);}}
    0 references
    DLMF id
    DLMF:14.3.E4
    0 references
    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Ferrers function of the first kind
    Defining formula
    𝖯 Ξ½ ΞΌ ⁑ ( x ) Ferrers-Legendre-P-first-kind πœ‡ 𝜈 π‘₯ {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2aadec
    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 π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2acdec
    0 references
    Q11589
    Defining formula
    x π‘₯ {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1cdec
    0 references
    Q11593
    Defining formula
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1dec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1bdec
    0 references
     
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