DLMF:30.8.E2 (Q8866)

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DLMF:30.8.E2
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    Statements

    Defining formula
    a n , k m ⁢ ( γ 2 ) = ( - 1 ) k ⁢ ( n + 2 ⁢ k + 1 2 ) ⁢ ( n - m + 2 ⁢ k ) ! ( n + m + 2 ⁢ k ) ! ⁢ ∫ - 1 1 𝖯𝗌 n m ⁡ ( x , γ 2 ) ⁢ 𝖯 n + 2 ⁢ k m ⁡ ( x ) ⁢ d x . subscript superscript 𝑎 𝑚 𝑛 𝑘 superscript 𝛾 2 superscript 1 𝑘 𝑛 2 𝑘 1 2 𝑛 𝑚 2 𝑘 𝑛 𝑚 2 𝑘 superscript subscript 1 1 spheroidal-wave-on-cut-Ps 𝑚 𝑛 𝑥 superscript 𝛾 2 Ferrers-Legendre-P-first-kind 𝑚 𝑛 2 𝑘 𝑥 𝑥 {\displaystyle{\displaystyle a^{m}_{n,k}(\gamma^{2})=(-1)^{k}\left(n+2k+\tfrac% {1}{2}\right)\frac{(n-m+2k)!}{(n+m+2k)!}\*\int_{-1}^{1}\mathsf{Ps}^{m}_{n}% \left(x,\gamma^{2}\right)\mathsf{P}^{m}_{n+2k}\left(x\right)\mathrm{d}x.}}
    0 references
    DLMF id
    DLMF:30.8.E2
    0 references
    Symbols used
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q12418
    Defining formula
    𝖯𝗌 n m ⁡ ( x , γ 2 ) spheroidal-wave-on-cut-Ps 𝑚 𝑛 𝑥 superscript 𝛾 2 {\displaystyle{\displaystyle\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},% \NVar{\gamma^{2}}\right)}}
    xml-id
    C30.S4.SS1.m1aadec
    0 references
    Q12419
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C30.S1.XMD2.m1adec
    0 references
    Q12408
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C30.S1.XMD4.m1adec
    0 references
    Q12409
    Defining formula
    n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}
    xml-id
    C30.S1.XMD5.m1adec
    0 references
    Q12404
    Defining formula
    γ 2 superscript 𝛾 2 {\displaystyle{\displaystyle\gamma^{2}}}
    xml-id
    C30.S2.XMD3.m1adec
    0 references
     
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