Formula:4539

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Digital Library of Mathematical Functions ID 10.65.E9

( ber ν x ) 2 + ( bei ν x ) 2 = ( 1 2 x ) 2 ν - 2 k = 0 2 k 2 + 2 ν k + 1 4 ν 2 Γ ( ν + k + 1 ) Γ ( ν + 2 k + 1 ) ( 1 4 x 2 ) 2 k k ! . superscript diffop Kelvin-ber 𝜈 1 𝑥 2 superscript diffop Kelvin-bei 𝜈 1 𝑥 2 superscript 1 2 𝑥 2 𝜈 2 superscript subscript 𝑘 0 2 superscript 𝑘 2 2 𝜈 𝑘 1 4 superscript 𝜈 2 Euler-Gamma 𝜈 𝑘 1 Euler-Gamma 𝜈 2 𝑘 1 superscript 1 4 superscript 𝑥 2 2 𝑘 𝑘 {\displaystyle{\displaystyle\left(\operatorname{ber}_{\nu}'x\right)^{2}+\left(% \operatorname{bei}_{\nu}'x\right)^{2}=(\tfrac{1}{2}x)^{2\nu-2}\sum_{k=0}^{% \infty}\frac{2k^{2}+2\nu k+\frac{1}{4}\nu^{2}}{\Gamma\left(\nu+k+1\right)% \Gamma\left(\nu+2k+1\right)}\frac{(\frac{1}{4}x^{2})^{2k}}{k!}.}}


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