DLMF:15.8.E10 (Q5067)

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DLMF:15.8.E10
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    Statements

    test
    𝐅 ⁑ ( a , b a + b + m ; z ) = 1 Ξ“ ⁑ ( a + m ) ⁒ Ξ“ ⁑ ( b + m ) ⁒ βˆ‘ k = 0 m - 1 ( a ) k ⁒ ( b ) k ⁒ ( m - k - 1 ) ! k ! ⁒ ( z - 1 ) k - ( z - 1 ) m Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( b ) ⁒ βˆ‘ 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 π‘Ž 𝑏 π‘Ž 𝑏 π‘š 𝑧 1 Euler-Gamma π‘Ž π‘š Euler-Gamma 𝑏 π‘š superscript subscript π‘˜ 0 π‘š 1 subscript π‘Ž π‘˜ subscript 𝑏 π‘˜ π‘š π‘˜ 1 π‘˜ superscript 𝑧 1 π‘˜ superscript 𝑧 1 π‘š Euler-Gamma π‘Ž Euler-Gamma 𝑏 superscript subscript π‘˜ 0 subscript π‘Ž π‘š π‘˜ subscript 𝑏 π‘š π‘˜ π‘˜ π‘˜ π‘š superscript 1 𝑧 π‘˜ 1 𝑧 digamma π‘˜ 1 digamma π‘˜ π‘š 1 digamma π‘Ž π‘˜ π‘š digamma 𝑏 π‘˜ π‘š {\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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    constraint
    | z - 1 | < 1 , | ph ⁑ ( 1 - z ) | < Ο€ formulae-sequence 𝑧 1 1 phase 1 𝑧 πœ‹ {\displaystyle{\displaystyle|z-1|<1,|\operatorname{ph}\left(1-z\right)|<\pi}}
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    Symbols used
    gamma function
    test
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
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    the ratio of the circumference of a circle to its diameter
    test
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    psi (or digamma) function
    test
    ψ ⁑ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
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    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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    principal branch of logarithm function
    test
    ln ⁑ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
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    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1agdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    test
    𝐅 ⁑ ( a , b ; c ; z ) scaled-hypergeometric-bold-F π‘Ž 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2agdec
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    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1idec
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    Q11645
    test
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1gdec
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    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1gdec
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    Q11658
    test
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C15.S1.XMD7.m1bdec
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    Q11651
    test
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1ddec
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