DLMF:14.19#Ex4 (Q4915)

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DLMF:14.19#Ex4
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    Defining formula
    P ν - 1 2 μ ⁡ ( cosh ⁡ ξ ) = Γ ⁡ ( 1 - 2 ⁢ μ ) ⁢ 2 2 ⁢ μ Γ ⁡ ( 1 - μ ) ⁢ ( 1 - e - 2 ⁢ ξ ) μ ⁢ e ( ν + ( 1 / 2 ) ) ⁢ ξ ⁢ 𝐅 ⁡ ( 1 2 - μ , 1 2 + ν - μ ; 1 - 2 ⁢ μ ; e - 2 ⁢ ξ ) , Legendre-P-first-kind 𝜇 𝜈 1 2 𝜉 Euler-Gamma 1 2 𝜇 superscript 2 2 𝜇 Euler-Gamma 1 𝜇 superscript 1 superscript 𝑒 2 𝜉 𝜇 superscript 𝑒 𝜈 1 2 𝜉 scaled-hypergeometric-bold-F 1 2 𝜇 1 2 𝜈 𝜇 1 2 𝜇 superscript 𝑒 2 𝜉 {\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=% \frac{\Gamma\left(1-2\mu\right)2^{2\mu}}{\Gamma\left(1-\mu\right)\left(1-e^{-2% \xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\mathbf{F}\left(\tfrac{1}{2}-\mu,\tfrac{% 1}{2}+\nu-\mu;1-2\mu;e^{-2\xi}\right),}}
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    DLMF id
    DLMF:14.19#Ex4
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