DLMF:14.3.E19 (Q4708)

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DLMF:14.3.E19
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    Statements

    Defining formula
    𝑸 Ξ½ ΞΌ ⁑ ( x ) = 2 Ξ½ ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) ⁒ ( x + 1 ) ΞΌ / 2 ( x - 1 ) ( ΞΌ / 2 ) + Ξ½ + 1 ⁒ 𝐅 ⁑ ( Ξ½ + 1 , Ξ½ + ΞΌ + 1 ; 2 ⁒ Ξ½ + 2 ; 2 1 - x ) , associated-Legendre-black-Q πœ‡ 𝜈 π‘₯ superscript 2 𝜈 Euler-Gamma 𝜈 1 superscript π‘₯ 1 πœ‡ 2 superscript π‘₯ 1 πœ‡ 2 𝜈 1 scaled-hypergeometric-bold-F 𝜈 1 𝜈 πœ‡ 1 2 𝜈 2 2 1 π‘₯ {\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
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2akdec
    0 references
    Q11592
    Defining formula
    𝑸 Ξ½ ΞΌ ⁑ ( z ) associated-Legendre-black-Q πœ‡ 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 π‘Ž 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 π‘₯ {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1rdec
    0 references
     
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