DLMF:14.19.E5 (Q4918)

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DLMF:14.19.E5
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    Statements

    Defining formula
    𝑸 n - 1 2 m ⁑ ( cosh ⁑ ΞΎ ) = Ξ“ ⁑ ( n + 1 2 ) Ξ“ ⁑ ( n + m + 1 2 ) ⁒ Ξ“ ⁑ ( n - m + 1 2 ) ⁒ ∫ 0 ∞ cosh ⁑ ( m ⁒ t ) ( cosh ⁑ ΞΎ + cosh ⁑ t ⁒ sinh ⁑ ΞΎ ) n + ( 1 / 2 ) ⁒ d t , associated-Legendre-black-Q π‘š 𝑛 1 2 πœ‰ Euler-Gamma 𝑛 1 2 Euler-Gamma 𝑛 π‘š 1 2 Euler-Gamma 𝑛 π‘š 1 2 superscript subscript 0 π‘š 𝑑 superscript πœ‰ 𝑑 πœ‰ 𝑛 1 2 𝑑 {\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi% \right)=\frac{\Gamma\left(n+\frac{1}{2}\right)}{\Gamma\left(n+m+\tfrac{1}{2}% \right)\Gamma\left(n-m+\frac{1}{2}\right)}\*\int_{0}^{\infty}\frac{\cosh\left(% mt\right)}{(\cosh\xi+\cosh t\sinh\xi)^{n+(1/2)}}\mathrm{d}t,}}
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    DLMF id
    DLMF:14.19.E5
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    constraint
    m < n + 1 2 π‘š 𝑛 1 2 {\displaystyle{\displaystyle m<n+\tfrac{1}{2}}}
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