DLMF:11.9.E8 (Q4016)

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DLMF:11.9.E8
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    Statements

    Defining formula
    s μ , ν ⁡ ( z ) = 2 ( μ + ν - 1 ) / 2 ⁢ Γ ⁡ ( 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 ) ⁢ z ( μ + 1 - ν ) / 2 ⁢ ∑ k = 0 ∞ ( 1 2 ⁢ z ) k k ! ⁢ ( 2 ⁢ k + μ - ν + 1 ) ⁢ J k + 1 2 ⁢ ( μ + ν + 1 ) ⁡ ( z ) . Lommel-s 𝜇 𝜈 𝑧 superscript 2 𝜇 𝜈 1 2 Euler-Gamma 1 2 𝜇 1 2 𝜈 1 2 superscript 𝑧 𝜇 1 𝜈 2 superscript subscript 𝑘 0 superscript 1 2 𝑧 𝑘 𝑘 2 𝑘 𝜇 𝜈 1 Bessel-J 𝑘 1 2 𝜇 𝜈 1 𝑧 {\displaystyle{\displaystyle s_{{\mu},{\nu}}\left(z\right)=2^{(\mu+\nu-1)/2}% \Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)z^{(\mu+1-\nu)/% 2}\*\sum_{k=0}^{\infty}\frac{(\tfrac{1}{2}z)^{k}}{k!(2k+\mu-\nu+1)}J_{k+\frac{% 1}{2}(\mu+\nu+1)}\left(z\right).}}
    0 references
    DLMF id
    DLMF:11.9.E8
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2acdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
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    Lommel function
    Defining formula
    s μ , ν ⁡ ( z ) Lommel-s 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right% )}}
    xml-id
    C11.S9.E3.m2aedec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
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    Q11491
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C11.S1.XMD2.m1fdec
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    Q11492
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C11.S1.XMD3.m1gdec
    0 references
    Q11497
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C11.S1.XMD5.m1cdec
    0 references
     
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