DLMF:30.8.E9 (Q8875)

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DLMF:30.8.E9
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    Statements

    Defining formula
    𝖰𝗌 n m ⁡ ( x , γ 2 ) = ∑ k = - ∞ - N - 1 ( - 1 ) k ⁢ a ′ n , k m ⁢ ( γ 2 ) ⁢ 𝖯 n + 2 ⁢ k m ⁡ ( x ) + ∑ k = - N ∞ ( - 1 ) k ⁢ a n , k m ⁢ ( γ 2 ) ⁢ 𝖰 n + 2 ⁢ k m ⁡ ( x ) , spheroidal-wave-on-cut-Qs 𝑚 𝑛 𝑥 superscript 𝛾 2 superscript subscript 𝑘 𝑁 1 superscript 1 𝑘 subscript superscript superscript 𝑎 ′ 𝑚 𝑛 𝑘 superscript 𝛾 2 Ferrers-Legendre-P-first-kind 𝑚 𝑛 2 𝑘 𝑥 superscript subscript 𝑘 𝑁 superscript 1 𝑘 subscript superscript 𝑎 𝑚 𝑛 𝑘 superscript 𝛾 2 Ferrers-Legendre-Q-first-kind 𝑚 𝑛 2 𝑘 𝑥 {\displaystyle{\displaystyle\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_% {k=-\infty}^{-N-1}(-1)^{k}{a^{\prime}}^{m}_{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+% 2k}\left(x\right)+\sum_{k=-N}^{\infty}(-1)^{k}a^{m}_{n,k}(\gamma^{2})\mathsf{Q% }^{m}_{n+2k}\left(x\right),}}
    0 references
    DLMF id
    DLMF:30.8.E9
    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.m2abdec
    0 references
    Ferrers function of the second kind
    Defining formula
    𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E2.m2adec
    0 references
    Q12421
    Defining formula
    𝖰𝗌 n m ⁡ ( x , γ 2 ) spheroidal-wave-on-cut-Qs 𝑚 𝑛 𝑥 superscript 𝛾 2 {\displaystyle{\displaystyle\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},% \NVar{\gamma^{2}}\right)}}
    xml-id
    C30.S5.m1adec
    0 references
    Q12419
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C30.S1.XMD2.m1bdec
    0 references
    Q12408
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C30.S1.XMD4.m1hdec
    0 references
    Q12409
    Defining formula
    n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}
    xml-id
    C30.S1.XMD5.m1hdec
    0 references
    Q12404
    Defining formula
    γ 2 superscript 𝛾 2 {\displaystyle{\displaystyle\gamma^{2}}}
    xml-id
    C30.S2.XMD3.m1hdec
    0 references
    coefficients (locally)
    Defining formula
    a n , k m ⁢ ( γ 2 ) subscript superscript 𝑎 𝑚 𝑛 𝑘 superscript 𝛾 2 {\displaystyle{\displaystyle a^{m}_{n,k}(\gamma^{2})}}
    xml-id
    C30.S8.XMD2.m1fdec
    0 references
    Q12428
    Defining formula
    N 𝑁 {\displaystyle{\displaystyle N}}
    xml-id
    C30.S8.XMD6.m1dec
    0 references
     
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