DLMF:15.8.E9 (Q5066): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11651 / rank
 
Normal rank
Property / Symbols used: Q11651 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11651 / qualifier
 
xml-id: C15.S1.XMD9.m1cdec

Latest revision as of 14:02, 2 January 2020

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DLMF:15.8.E9
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    Statements

    Defining formula
    𝐅 ⁡ ( a , a + m c ; z ) = ( 1 - z ) - a Γ ⁡ ( a + m ) ⁢ Γ ⁡ ( c - a ) ⁢ ∑ k = 0 m - 1 ( a ) k ⁢ ( c - a - m ) k ⁢ ( m - k - 1 ) ! k ! ⁢ ( z - 1 ) - k + ( - 1 ) m ⁢ ( 1 - z ) - a - m Γ ⁡ ( a ) ⁢ Γ ⁡ ( c - a - m ) ⁢ ∑ 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 𝑎 𝑎 𝑚 𝑐 𝑧 superscript 1 𝑧 𝑎 Euler-Gamma 𝑎 𝑚 Euler-Gamma 𝑐 𝑎 superscript subscript 𝑘 0 𝑚 1 subscript 𝑎 𝑘 subscript 𝑐 𝑎 𝑚 𝑘 𝑚 𝑘 1 𝑘 superscript 𝑧 1 𝑘 superscript 1 𝑚 superscript 1 𝑧 𝑎 𝑚 Euler-Gamma 𝑎 Euler-Gamma 𝑐 𝑎 𝑚 superscript subscript 𝑘 0 subscript 𝑎 𝑚 𝑘 subscript 𝑐 𝑎 𝑘 𝑘 𝑘 𝑚 superscript 1 𝑧 𝑘 1 𝑧 digamma 𝑘 1 digamma 𝑘 𝑚 1 digamma 𝑎 𝑘 𝑚 digamma 𝑐 𝑎 𝑘 {\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)),}}
    0 references
    DLMF id
    DLMF:15.8.E9
    0 references
    constraint
    | z - 1 | > 1 , | ph ⁡ ( 1 - z ) | < π formulae-sequence 𝑧 1 1 phase 1 𝑧 𝜋 {\displaystyle{\displaystyle|z-1|>1,|\operatorname{ph}\left(1-z\right)|<\pi}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q10817
    Defining formula
    ph phase {\displaystyle{\displaystyle\operatorname{ph}}}
    xml-id
    C1.S9.E7.m1afdec
    0 references
    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2afdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1hdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\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
    0 references
    Q11658
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C15.S1.XMD7.m1adec
    0 references
    Q11651
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1cdec
    0 references
     
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