DLMF:10.44.E6 (Q3654)

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DLMF:10.44.E6
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    K n ⁡ ( z ) = n ! ⁢ ( 1 2 ⁢ z ) - n 2 ⁢ ∑ k = 0 n - 1 ( - 1 ) k ⁢ ( 1 2 ⁢ z ) k ⁢ I k ⁡ ( z ) k ! ⁢ ( n - k ) + ( - 1 ) n - 1 ⁢ ( ln ⁡ ( 1 2 ⁢ z ) - ψ ⁡ ( n + 1 ) ) ⁢ I n ⁡ ( z ) + ( - 1 ) n ⁢ ∑ k = 1 ∞ ( n + 2 ⁢ k ) ⁢ I n + 2 ⁢ k ⁡ ( z ) k ⁢ ( n + k ) , modified-Bessel-second-kind 𝑛 𝑧 𝑛 superscript 1 2 𝑧 𝑛 2 superscript subscript 𝑘 0 𝑛 1 superscript 1 𝑘 superscript 1 2 𝑧 𝑘 modified-Bessel-first-kind 𝑘 𝑧 𝑘 𝑛 𝑘 superscript 1 𝑛 1 1 2 𝑧 digamma 𝑛 1 modified-Bessel-first-kind 𝑛 𝑧 superscript 1 𝑛 superscript subscript 𝑘 1 𝑛 2 𝑘 modified-Bessel-first-kind 𝑛 2 𝑘 𝑧 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle K_{n}\left(z\right)=\frac{n!(\tfrac{1}{2}z)^{-n}}% {2}\sum_{k=0}^{n-1}(-1)^{k}\frac{(\tfrac{1}{2}z)^{k}I_{k}\left(z\right)}{k!(n-% k)}+(-1)^{n-1}\left(\ln\left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)I% _{n}\left(z\right)+(-1)^{n}\sum_{k=1}^{\infty}\frac{(n+2k)I_{n+2k}\left(z% \right)}{k(n+k)},}}
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    DLMF:10.44.E6
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