DLMF:14.3.E19 (Q4708): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11591 / rank
 
Normal rank
Property / Symbols used: Q11591 / qualifier
 
Defining formula:

μ 𝜇 {\displaystyle{\displaystyle\mu}}

\mu
Property / Symbols used: Q11591 / qualifier
 
xml-id: C14.S1.XMD7.m1ndec

Revision as of 01:19, 2 January 2020

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DLMF:14.3.E19
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    Statements

    Defining formula
    𝑸 ν μ ⁡ ( x ) = 2 ν ⁢ Γ ⁡ ( ν + 1 ) ⁢ ( x + 1 ) μ / 2 ( x - 1 ) ( μ / 2 ) + ν + 1 ⁢ 𝐅 ⁡ ( ν + 1 , ν + μ + 1 ; 2 ⁢ ν + 2 ; 2 1 - x ) , associated-Legendre-black-Q 𝜇 𝜈 𝑥 superscript 2 𝜈 Euler-Gamma 𝜈 1 superscript 𝑥 1 𝜇 2 superscript 𝑥 1 𝜇 2 𝜈 1 scaled-hypergeometric-bold-F 𝜈 1 𝜈 𝜇 1 2 𝜈 2 2 1 𝑥 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{2^{% \nu}\Gamma\left(\nu+1\right)(x+1)^{\mu/2}}{(x-1)^{(\mu/2)+\nu+1}}\mathbf{F}% \left(\nu+1,\nu+\mu+1;2\nu+2;\frac{2}{1-x}\right),}}
    0 references
    DLMF id
    DLMF:14.3.E19
    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.m2akdec
    0 references
    Q11592
    Defining formula
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3aadec
    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.m2aodec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1rdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1ndec
    0 references
     
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