DLMF:28.24.E7 (Q8404)

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DLMF:28.24.E7
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    Statements

    Defining formula
    Io 2 ⁢ m + 2 ⁡ ( z , h ) = ( - 1 ) s ⁢ ∑ ℓ = 0 ∞ ( - 1 ) ℓ ⁢ B 2 ⁢ ℓ + 2 2 ⁢ m + 2 ⁢ ( h 2 ) B 2 ⁢ s + 2 2 ⁢ m + 2 ⁢ ( h 2 ) ⁢ ( I ℓ - s ⁡ ( h ⁢ e - z ) ⁢ I ℓ + s + 2 ⁡ ( h ⁢ e z ) - I ℓ + s + 2 ⁡ ( h ⁢ e - z ) ⁢ I ℓ - s ⁡ ( h ⁢ e z ) ) , modified-Mathieu-Io 2 𝑚 2 𝑧 ℎ superscript 1 𝑠 superscript subscript ℓ 0 superscript 1 ℓ superscript subscript 𝐵 2 ℓ 2 2 𝑚 2 superscript ℎ 2 superscript subscript 𝐵 2 𝑠 2 2 𝑚 2 superscript ℎ 2 modified-Bessel-first-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 2 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 2 ℎ superscript 𝑒 𝑧 modified-Bessel-first-kind ℓ 𝑠 ℎ superscript 𝑒 𝑧 {\displaystyle{\displaystyle\mathrm{Io}_{2m+2}\left(z,h\right)=(-1)^{s}\sum_{% \ell=0}^{\infty}(-1)^{\ell}\dfrac{B_{2\ell+2}^{2m+2}(h^{2})}{B_{2s+2}^{2m+2}(h% ^{2})}\left(I_{\ell-s}\left(he^{-z}\right)I_{\ell+s+2}\left(he^{z}\right)-I_{% \ell+s+2}\left(he^{-z}\right)I_{\ell-s}\left(he^{z}\right)\right),}}
    0 references
    DLMF id
    DLMF:28.24.E7
    0 references
    Symbols used
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
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    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2aadec
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    modified Mathieu function
    Defining formula
    Io n ⁡ ( z , h ) modified-Mathieu-Io 𝑛 𝑧 ℎ {\displaystyle{\displaystyle\mathrm{Io}_{\NVar{n}}\left(\NVar{z},\NVar{h}% \right)}}
    xml-id
    C28.S20.E18.m2adec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1edec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1fdec
    0 references
    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1fdec
    0 references
    Q12288
    Defining formula
    B m ⁢ ( q ) subscript 𝐵 𝑚 𝑞 {\displaystyle{\displaystyle B_{m}(q)}}
    xml-id
    C28.S4.XMD2.m1bdec
    0 references
     
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