DLMF:15.8.E11 (Q5068)

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DLMF:15.8.E11
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    Statements

    Defining formula
    𝐅 ⁡ ( a , b a + b + m ; z ) = z - a Γ ⁡ ( a + m ) ⁢ ∑ k = 0 m - 1 ( a ) k ⁢ ( m - k - 1 ) ! k ! ⁢ Γ ⁡ ( b + m - k ) ⁢ ( 1 - 1 z ) k - z - a Γ ⁡ ( a ) ⁢ ∑ k = 0 ∞ ( a + m ) k k ! ⁢ ( k + m ) ! ⁢ Γ ⁡ ( b - k ) ⁢ ( - 1 ) k ⁢ ( 1 - 1 z ) k + m ⁢ ( ln ⁡ ( 1 - z z ) - ψ ⁡ ( k + 1 ) - ψ ⁡ ( k + m + 1 ) + ψ ⁡ ( a + k + m ) + ψ ⁡ ( b - k ) ) , scaled-hypergeometric-bold-F 𝑎 𝑏 𝑎 𝑏 𝑚 𝑧 superscript 𝑧 𝑎 Euler-Gamma 𝑎 𝑚 superscript subscript 𝑘 0 𝑚 1 subscript 𝑎 𝑘 𝑚 𝑘 1 𝑘 Euler-Gamma 𝑏 𝑚 𝑘 superscript 1 1 𝑧 𝑘 superscript 𝑧 𝑎 Euler-Gamma 𝑎 superscript subscript 𝑘 0 subscript 𝑎 𝑚 𝑘 𝑘 𝑘 𝑚 Euler-Gamma 𝑏 𝑘 superscript 1 𝑘 superscript 1 1 𝑧 𝑘 𝑚 1 𝑧 𝑧 digamma 𝑘 1 digamma 𝑘 𝑚 1 digamma 𝑎 𝑘 𝑚 digamma 𝑏 𝑘 {\displaystyle{\displaystyle\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{z^% {-a}}{\Gamma\left(a+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\Gamma% \left(b+m-k\right)}\left(1-\frac{1}{z}\right)^{k}-\frac{z^{-a}}{\Gamma\left(a% \right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}}{k!(k+m)!\Gamma\left(b-k\right)}(-1% )^{k}\left(1-\frac{1}{z}\right)^{k+m}\*\left(\ln\left(\frac{1-z}{z}\right)-% \psi\left(k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b% -k\right)\right),}}
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    DLMF id
    DLMF:15.8.E11
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    constraint
    ℜ ⁡ z > 1 2 , | ph ⁡ z | < π , | ph ⁡ ( 1 - z ) | < π formulae-sequence 𝑧 1 2 formulae-sequence phase 𝑧 𝜋 phase 1 𝑧 𝜋 {\displaystyle{\displaystyle\Re z>\tfrac{1}{2},|\operatorname{ph}z|<\pi,|% \operatorname{ph}\left(1-z\right)|<\pi}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
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