DLMF:15.8.E9 (Q5066)

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DLMF:15.8.E9
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    Statements

    Defining formula
    𝐅 ⁑ ( a , a + m c ; z ) = ( 1 - z ) - a Ξ“ ⁑ ( a + m ) ⁒ Ξ“ ⁑ ( c - a ) ⁒ βˆ‘ k = 0 m - 1 ( a ) k ⁒ ( c - a - m ) k ⁒ ( m - k - 1 ) ! k ! ⁒ ( z - 1 ) - k + ( - 1 ) m ⁒ ( 1 - z ) - a - m Ξ“ ⁑ ( a ) ⁒ Ξ“ ⁑ ( c - a - m ) ⁒ βˆ‘ k = 0 ∞ ( a + m ) k ⁒ ( c - a ) k k ! ⁒ ( k + m ) ! ⁒ ( 1 - z ) - k ⁒ ( ln ⁑ ( 1 - z ) + ψ ⁑ ( k + 1 ) + ψ ⁑ ( k + m + 1 ) - ψ ⁑ ( a + k + m ) - ψ ⁑ ( c - a + k ) ) , scaled-hypergeometric-bold-F π‘Ž π‘Ž π‘š 𝑐 𝑧 superscript 1 𝑧 π‘Ž Euler-Gamma π‘Ž π‘š Euler-Gamma 𝑐 π‘Ž superscript subscript π‘˜ 0 π‘š 1 subscript π‘Ž π‘˜ subscript 𝑐 π‘Ž π‘š π‘˜ π‘š π‘˜ 1 π‘˜ superscript 𝑧 1 π‘˜ superscript 1 π‘š superscript 1 𝑧 π‘Ž π‘š Euler-Gamma π‘Ž Euler-Gamma 𝑐 π‘Ž π‘š superscript subscript π‘˜ 0 subscript π‘Ž π‘š π‘˜ subscript 𝑐 π‘Ž 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 𝑐 π‘Ž π‘˜ {\displaystyle{\displaystyle\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(1-z% )^{-a}}{\Gamma\left(a+m\right)\Gamma\left(c-a\right)}\sum_{k=0}^{m-1}\frac{(a)% _{k}(c-a-m)_{k}(m-k-1)!}{k!}(z-1)^{-k}+\frac{(-1)^{m}(1-z)^{-a-m}}{\Gamma\left% (a\right)\Gamma\left(c-a-m\right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(c-a)_{k}}% {k!(k+m)!}(1-z)^{-k}\*(\ln\left(1-z\right)+\psi\left(k+1\right)+\psi\left(k+m+% 1\right)-\psi\left(a+k+m\right)-\psi\left(c-a+k\right)),}}
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    DLMF id
    DLMF:15.8.E9
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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
    Defining formula
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
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    psi (or digamma) function
    Defining formula
    ψ ⁑ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
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    principal branch of logarithm function
    Defining formula
    ln ⁑ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
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    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1afdec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    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.m2afdec
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    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1hdec
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    Q11645
    Defining formula
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1fdec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1hdec
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