DLMF:14.18.E4 (Q4905)

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DLMF:14.18.E4
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    Statements

    Defining formula
    P ν ⁡ ( cosh ⁡ ξ 1 ⁢ cosh ⁡ ξ 2 - sinh ⁡ ξ 1 ⁢ sinh ⁡ ξ 2 ⁢ cos ⁡ ϕ ) = P ν ⁡ ( cosh ⁡ ξ 1 ) ⁢ P ν ⁡ ( cosh ⁡ ξ 2 ) + 2 ⁢ ∑ m = 1 ∞ ( - 1 ) m ⁢ P ν - m ⁡ ( cosh ⁡ ξ 1 ) ⁢ P ν m ⁡ ( cosh ⁡ ξ 2 ) ⁢ cos ⁡ ( m ⁢ ϕ ) , shorthand-Legendre-P-first-kind 𝜈 subscript 𝜉 1 subscript 𝜉 2 subscript 𝜉 1 subscript 𝜉 2 italic-ϕ shorthand-Legendre-P-first-kind 𝜈 subscript 𝜉 1 shorthand-Legendre-P-first-kind 𝜈 subscript 𝜉 2 2 superscript subscript 𝑚 1 superscript 1 𝑚 Legendre-P-first-kind 𝑚 𝜈 subscript 𝜉 1 Legendre-P-first-kind 𝑚 𝜈 subscript 𝜉 2 𝑚 italic-ϕ {\displaystyle{\displaystyle P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1% }\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(% \cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}% \right)P^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right),}}
    0 references
    DLMF id
    DLMF:14.18.E4
    0 references
    Symbols used
    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1adec
    0 references
     
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