DLMF:10.65.E7 (Q3840)

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DLMF:10.65.E7
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    ber ν ⁡ x ⁢ bei ν ′ ⁡ x - ber ν ′ ⁡ x ⁢ bei ν ⁡ x = ( 1 2 ⁢ x ) 2 ⁢ ν + 1 ⁢ ∑ k = 0 ∞ 1 Γ ⁡ ( ν + k + 1 ) ⁢ Γ ⁡ ( ν + 2 ⁢ k + 2 ) ⁢ ( 1 4 ⁢ x 2 ) 2 ⁢ k k ! , Kelvin-ber 𝜈 𝑥 diffop Kelvin-bei 𝜈 1 𝑥 diffop Kelvin-ber 𝜈 1 𝑥 Kelvin-bei 𝜈 𝑥 superscript 1 2 𝑥 2 𝜈 1 superscript subscript 𝑘 0 1 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜈 2 𝑘 2 superscript 1 4 superscript 𝑥 2 2 𝑘 𝑘 {\displaystyle{\displaystyle\operatorname{ber}_{\nu}x\operatorname{bei}_{\nu}'% x-\operatorname{ber}_{\nu}'x\operatorname{bei}_{\nu}x=(\tfrac{1}{2}x)^{2\nu+1}% \sum_{k=0}^{\infty}\frac{1}{\Gamma\left(\nu+k+1\right)\Gamma\left(\nu+2k+2% \right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!},}}
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    DLMF:10.65.E7
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