DLMF:14.3.E8 (Q4697): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
 
Normal rank
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
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)}}

\hyperOlverF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
 
xml-id: C15.S2.E2.m2agdec

Revision as of 01:14, 2 January 2020

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DLMF:14.3.E8
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    Statements

    Defining formula
    P ν m ⁡ ( x ) = Γ ⁡ ( ν + m + 1 ) 2 m ⁢ Γ ⁡ ( ν - m + 1 ) ⁢ ( x 2 - 1 ) m / 2 ⁢ 𝐅 ⁡ ( ν + m + 1 , m - ν ; m + 1 ; 1 2 - 1 2 ⁢ x ) . Legendre-P-first-kind 𝑚 𝜈 𝑥 Euler-Gamma 𝜈 𝑚 1 superscript 2 𝑚 Euler-Gamma 𝜈 𝑚 1 superscript superscript 𝑥 2 1 𝑚 2 scaled-hypergeometric-bold-F 𝜈 𝑚 1 𝑚 𝜈 𝑚 1 1 2 1 2 𝑥 {\displaystyle{\displaystyle P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m% +1\right)}{2^{m}\Gamma\left(\nu-m+1\right)}\left(x^{2}-1\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.E8
    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.m2aedec
    0 references
    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.m1aadec
    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.m2agdec
    0 references
     
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