DLMF:14.3.E4 (Q4693): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1cdec

Revision as of 01:13, 2 January 2020

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DLMF:14.3.E4
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    Statements

    Defining formula
    𝖯 ν m ⁡ ( x ) = ( - 1 ) m ⁢ Γ ⁡ ( ν + m + 1 ) 2 m ⁢ Γ ⁡ ( ν - m + 1 ) ⁢ ( 1 - x 2 ) m / 2 ⁢ 𝐅 ⁡ ( ν + m + 1 , m - ν ; m + 1 ; 1 2 - 1 2 ⁢ x ) ; Ferrers-Legendre-P-first-kind 𝑚 𝜈 𝑥 superscript 1 𝑚 Euler-Gamma 𝜈 𝑚 1 superscript 2 𝑚 Euler-Gamma 𝜈 𝑚 1 superscript 1 superscript 𝑥 2 𝑚 2 scaled-hypergeometric-bold-F 𝜈 𝑚 1 𝑚 𝜈 𝑚 1 1 2 1 2 𝑥 {\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
    Γ ⁡ ( 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 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\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 𝑎 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 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1cdec
    0 references
     
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