DLMF:14.19.E7 (Q4920)

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DLMF:14.19.E7
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    Statements

    Defining formula
    P n - 1 2 m ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( n + m + 1 2 ) Γ ⁡ ( n - m + 1 2 ) ⁢ ( 2 π ⁢ sinh ⁡ ξ ) 1 / 2 ⁢ 𝑸 m - 1 2 n ⁡ ( coth ⁡ ξ ) , Legendre-P-first-kind 𝑚 𝑛 1 2 𝜉 Euler-Gamma 𝑛 𝑚 1 2 Euler-Gamma 𝑛 𝑚 1 2 superscript 2 𝜋 𝜉 1 2 associated-Legendre-black-Q 𝑛 𝑚 1 2 hyperbolic-cotangent 𝜉 {\displaystyle{\displaystyle P^{m}_{n-\frac{1}{2}}\left(\cosh\xi\right)=\frac{% \Gamma\left(n+m+\tfrac{1}{2}\right)}{\Gamma\left(n-m+\tfrac{1}{2}\right)}\*% \left(\frac{2}{\pi\sinh\xi}\right)^{1/2}\boldsymbol{Q}^{n}_{m-\frac{1}{2}}% \left(\coth\xi\right),}}
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    DLMF id
    DLMF:14.19.E7
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