DLMF:10.60.E4 (Q3779)

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DLMF:10.60.E4
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    Statements

    Defining formula
    𝗃 n ⁡ ( 2 ⁢ z ) = - n ! ⁢ z n + 1 ⁢ ∑ k = 0 n 2 ⁢ n - 2 ⁢ k + 1 k ! ⁢ ( 2 ⁢ n - k + 1 ) ! ⁢ 𝗃 n - k ⁡ ( z ) ⁢ 𝗒 n - k ⁡ ( z ) , spherical-Bessel-J 𝑛 2 𝑧 𝑛 superscript 𝑧 𝑛 1 superscript subscript 𝑘 0 𝑛 2 𝑛 2 𝑘 1 𝑘 2 𝑛 𝑘 1 spherical-Bessel-J 𝑛 𝑘 𝑧 spherical-Bessel-Y 𝑛 𝑘 𝑧 {\displaystyle{\displaystyle\mathsf{j}_{n}\left(2z\right)=-n!z^{n+1}\sum_{k=0}% ^{n}\frac{2n-2k+1}{k!(2n-k+1)!}\mathsf{j}_{n-k}\left(z\right)\mathsf{y}_{n-k}% \left(z\right),}}
    0 references
    DLMF id
    DLMF:10.60.E4
    0 references
    Symbols used
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    spherical Bessel function of the first kind
    Defining formula
    𝗃 n ⁡ ( z ) spherical-Bessel-J 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C10.S47.E3.m2abdec
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    spherical Bessel function of the second kind
    Defining formula
    𝗒 n ⁡ ( z ) spherical-Bessel-Y 𝑛 𝑧 {\displaystyle{\displaystyle\mathsf{y}_{\NVar{n}}\left(\NVar{z}\right)}}
    xml-id
    C10.S47.E4.m2aadec
    0 references
    Q11429
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C10.S1.XMD2.m1cdec
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    Q11428
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1dec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1dec
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