DLMF:14.19.E2 (Q4914)

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DLMF:14.19.E2
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    P ν - 1 2 μ ( cosh ξ ) = Γ ( 1 2 - μ ) π 1 / 2 ( 1 - e - 2 ξ ) μ e ( ν + ( 1 / 2 ) ) ξ 𝐅 ( 1 2 - μ , 1 2 + ν - μ ; 1 - 2 μ ; 1 - e - 2 ξ ) , Legendre-P-first-kind 𝜇 𝜈 1 2 𝜉 Euler-Gamma 1 2 𝜇 superscript 𝜋 1 2 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 1 superscript 𝑒 2 𝜉 {\displaystyle{\displaystyle P^{\mu}_{\nu-\frac{1}{2}}\left(\cosh\xi\right)=% \frac{\Gamma\left(\frac{1}{2}-\mu\right)}{\pi^{1/2}\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;1-e^{-2\xi}\right),}}
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    DLMF:14.19.E2
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    μ 1 2 , 3 2 , 5 2 , 𝜇 1 2 3 2 5 2 {\displaystyle{\displaystyle\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots}}
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