DLMF:14.19.E8 (Q4921)

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DLMF:14.19.E8
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    Statements

    Defining formula
    𝑸 n - 1 2 m ⁑ ( cosh ⁑ ΞΎ ) = Ξ“ ⁑ ( m - n + 1 2 ) Ξ“ ⁑ ( m + n + 1 2 ) ⁒ ( Ο€ 2 ⁒ sinh ⁑ ΞΎ ) 1 / 2 ⁒ P m - 1 2 n ⁑ ( coth ⁑ ΞΎ ) . associated-Legendre-black-Q π‘š 𝑛 1 2 πœ‰ Euler-Gamma π‘š 𝑛 1 2 Euler-Gamma π‘š 𝑛 1 2 superscript πœ‹ 2 πœ‰ 1 2 Legendre-P-first-kind 𝑛 π‘š 1 2 hyperbolic-cotangent πœ‰ {\displaystyle{\displaystyle\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\left(\cosh\xi% \right)=\frac{\Gamma\left(m-n+\tfrac{1}{2}\right)}{\Gamma\left(m+n+\tfrac{1}{2% }\right)}\*\left(\frac{\pi}{2\sinh\xi}\right)^{1/2}P^{n}_{m-\frac{1}{2}}\left(% \coth\xi\right).}}
    0 references
    DLMF id
    DLMF:14.19.E8
    0 references
    Symbols used
    gamma function
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    Q11481
    Defining formula
    P Ξ½ ΞΌ ⁑ ( z ) Legendre-P-first-kind πœ‡ 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1acdec
    0 references
    Q11592
    Defining formula
    𝑸 Ξ½ ΞΌ ⁑ ( z ) associated-Legendre-black-Q πœ‡ 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁑ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2agdec
    0 references
     
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