DLMF:10.31.E3 (Q3520)

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DLMF:10.31.E3
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    Statements

    test
    I ν ⁡ ( z ) ⁢ I μ ⁡ ( z ) = ( 1 2 ⁢ z ) ν + μ ⁢ ∑ k = 0 ∞ ( ν + μ + k + 1 ) k ⁢ ( 1 4 ⁢ z 2 ) k k ! ⁢ Γ ⁡ ( ν + k + 1 ) ⁢ Γ ⁡ ( μ + k + 1 ) . modified-Bessel-first-kind 𝜈 𝑧 modified-Bessel-first-kind 𝜇 𝑧 superscript 1 2 𝑧 𝜈 𝜇 superscript subscript 𝑘 0 subscript 𝜈 𝜇 𝑘 1 𝑘 superscript 1 4 superscript 𝑧 2 𝑘 𝑘 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜇 𝑘 1 {\displaystyle{\displaystyle I_{\nu}\left(z\right)I_{\mu}\left(z\right)=(% \tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(\tfrac{1}{4% }z^{2})^{k}}{k!\Gamma\left(\nu+k+1\right)\Gamma\left(\mu+k+1\right)}.}}
    0 references
    DLMF id
    DLMF:10.31.E3
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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    modified Bessel function of the first kind
    test
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2abdec
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    Q11428
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C10.S1.XMD3.m1adec
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    Q11426
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1bdec
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    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1dec
    0 references
     
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