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DLMF:14.7.E14 - MaRDI portal

DLMF:14.7.E14 (Q5463)

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DLMF:14.7.E14
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    Statements

    DLMF defining formula
    P n m ⁡ ( x ) = ( x 2 - 1 ) m / 2 2 n ⁢ n ! ⁢ d m + n d x m + n ⁡ ( x 2 - 1 ) n , Legendre-P-first-kind 𝑚 𝑛 𝑥 superscript superscript 𝑥 2 1 𝑚 2 superscript 2 𝑛 𝑛 derivative 𝑥 𝑚 𝑛 superscript superscript 𝑥 2 1 𝑛 {\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right% )^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(x^{2}-1% \right)^{n},}}
    0 references
    Symbols used
    associated Legendre function of the first kind
    DLMF 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.m1abdec
    0 references
    derivative of $$f$$ with respect to $$x$$
    DLMF defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2afdec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1mdec
    0 references
    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1edec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1ldec
    0 references

    Identifiers

     
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