DLMF:15.8.E10 (Q5067)

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DLMF:15.8.E10
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    test
    𝐅 ⁡ ( a , b a + b + m ; z ) = 1 Γ ⁡ ( a + m ) ⁢ Γ ⁡ ( b + m ) ⁢ ∑ k = 0 m - 1 ( a ) k ⁢ ( b ) k ⁢ ( m - k - 1 ) ! k ! ⁢ ( z - 1 ) k - ( z - 1 ) m Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ ∑ k = 0 ∞ ( a + m ) k ⁢ ( b + m ) k k ! ⁢ ( k + m ) ! ⁢ ( 1 - z ) k ⁢ ( ln ⁡ ( 1 - z ) - ψ ⁡ ( k + 1 ) - ψ ⁡ ( k + m + 1 ) + ψ ⁡ ( a + k + m ) + ψ ⁡ ( b + k + m ) ) , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑎 𝑏 𝑚 𝑧 1 Euler-Gamma 𝑎 𝑚 Euler-Gamma 𝑏 𝑚 superscript subscript 𝑘 0 𝑚 1 subscript 𝑎 𝑘 subscript 𝑏 𝑘 𝑚 𝑘 1 𝑘 superscript 𝑧 1 𝑘 superscript 𝑧 1 𝑚 Euler-Gamma 𝑎 Euler-Gamma 𝑏 superscript subscript 𝑘 0 subscript 𝑎 𝑚 𝑘 subscript 𝑏 𝑚 𝑘 𝑘 𝑘 𝑚 superscript 1 𝑧 𝑘 1 𝑧 digamma 𝑘 1 digamma 𝑘 𝑚 1 digamma 𝑎 𝑘 𝑚 digamma 𝑏 𝑘 𝑚 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{1}% {\Gamma\left(a+m\right)\Gamma\left(b+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(b)% _{k}(m-k-1)!}{k!}(z-1)^{k}-\frac{(z-1)^{m}}{\Gamma\left(a\right)\Gamma\left(b% \right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(b+m)_{k}}{k!(k+m)!}(1-z)^{k}\*\left% (\ln\left(1-z\right)-\psi\left(k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k% +m\right)+\psi\left(b+k+m\right)\right),}}
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    DLMF id
    DLMF:15.8.E10
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