DLMF:15.8.E11 (Q5068)

From testwiki
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:15.8.E11
No description defined

    Statements

    test
    𝐅 ⁑ ( a , b a + b + m ; z ) = z - a Ξ“ ⁑ ( a + m ) ⁒ βˆ‘ k = 0 m - 1 ( a ) k ⁒ ( m - k - 1 ) ! k ! ⁒ Ξ“ ⁑ ( b + m - k ) ⁒ ( 1 - 1 z ) k - z - a Ξ“ ⁑ ( a ) ⁒ βˆ‘ k = 0 ∞ ( a + m ) k k ! ⁒ ( k + m ) ! ⁒ Ξ“ ⁑ ( 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 π‘Ž 𝑏 π‘Ž 𝑏 π‘š 𝑧 superscript 𝑧 π‘Ž Euler-Gamma π‘Ž π‘š superscript subscript π‘˜ 0 π‘š 1 subscript π‘Ž π‘˜ π‘š π‘˜ 1 π‘˜ Euler-Gamma 𝑏 π‘š π‘˜ superscript 1 1 𝑧 π‘˜ superscript 𝑧 π‘Ž Euler-Gamma π‘Ž superscript subscript π‘˜ 0 subscript π‘Ž π‘š π‘˜ π‘˜ π‘˜ π‘š Euler-Gamma 𝑏 π‘˜ superscript 1 π‘˜ superscript 1 1 𝑧 π‘˜ π‘š 1 𝑧 𝑧 digamma π‘˜ 1 digamma π‘˜ π‘š 1 digamma π‘Ž π‘˜ π‘š digamma 𝑏 π‘˜ {\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),}}
    0 references
    DLMF id
    DLMF:15.8.E11
    0 references
    constraint
    β„œ ⁑ z > 1 2 , | ph ⁑ z | < Ο€ , | ph ⁑ ( 1 - z ) | < Ο€ formulae-sequence 𝑧 1 2 formulae-sequence phase 𝑧 πœ‹ phase 1 𝑧 πœ‹ {\displaystyle{\displaystyle\Re z>\tfrac{1}{2},|\operatorname{ph}z|<\pi,|% \operatorname{ph}\left(1-z\right)|<\pi}}
    0 references
    Symbols used
    gamma function
    test
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    Ο€ {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    psi (or digamma) function
    test
    ψ ⁑ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2acdec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
    principal branch of logarithm function
    test
    ln ⁑ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2acdec
    0 references
    Q10817
    test
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1ahdec
    0 references
    Q10811
    test
    β„œ ⁑ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ 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.m2ahdec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1jdec
    0 references
    Q11645
    test
    a π‘Ž {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1hdec
    0 references
    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1hdec
    0 references
    Q11658
    test
    k π‘˜ {\displaystyle{\displaystyle k}}
    xml-id
    C15.S1.XMD7.m1cdec
    0 references
    Q11651
    test
    m π‘š {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1edec
    0 references
     
      Retrieved from "https://lct.wmflabs.org/w/index.php?title=Item:Q5068&oldid=28929"