DLMF:15.10.E8 (Q5124)

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DLMF:15.10.E8
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    test
    F ⁡ ( a , b n ; z ) ⁢ ln ⁡ z - ∑ k = 1 n - 1 ( n - 1 ) ! ⁢ ( k - 1 ) ! ( n - k - 1 ) ! ⁢ ( 1 - a ) k ⁢ ( 1 - b ) k ⁢ ( - z ) - k + ∑ k = 0 ∞ ( a ) k ⁢ ( b ) k ( n ) k ⁢ k ! ⁢ z k ⁢ ( ψ ⁡ ( a + k ) + ψ ⁡ ( b + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( n + k ) ) , Gauss-hypergeometric-F 𝑎 𝑏 𝑛 𝑧 𝑧 superscript subscript 𝑘 1 𝑛 1 𝑛 1 𝑘 1 𝑛 𝑘 1 Pochhammer 1 𝑎 𝑘 Pochhammer 1 𝑏 𝑘 superscript 𝑧 𝑘 superscript subscript 𝑘 0 Pochhammer 𝑎 𝑘 Pochhammer 𝑏 𝑘 Pochhammer 𝑛 𝑘 𝑘 superscript 𝑧 𝑘 digamma 𝑎 𝑘 digamma 𝑏 𝑘 digamma 1 𝑘 digamma 𝑛 𝑘 {\displaystyle{\displaystyle F\left({a,b\atop n};z\right)\ln z-\sum_{k=1}^{n-1% }\frac{(n-1)!(k-1)!}{(n-k-1)!{\left(1-a\right)_{k}}{\left(1-b\right)_{k}}}(-z)% ^{-k}\\ +\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b\right)_{k}}}{{\left(n% \right)_{k}}k!}z^{k}\left(\psi\left(a+k\right)+\psi\left(b+k\right)-\psi\left(% 1+k\right)-\psi\left(n+k\right)\right),}}
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    DLMF:15.10.E8
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