DLMF:15.8.E8 (Q5065): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2addec

Revision as of 13:02, 2 January 2020

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DLMF:15.8.E8
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    Statements

    test
    𝐅 ⁡ ( a , a + m c ; z ) = ( - z ) - a Γ ⁡ ( a + m ) ⁢ ∑ k = 0 m - 1 ( a ) k ⁢ ( m - k - 1 ) ! k ! ⁢ Γ ⁡ ( c - a - k ) ⁢ z - k + ( - z ) - a Γ ⁡ ( a ) ⁢ ∑ k = 0 ∞ ( a + m ) k k ! ⁢ ( k + m ) ! ⁢ Γ ⁡ ( c - a - k - m ) ⁢ ( - 1 ) k ⁢ z - k - m ⁢ ( ln ⁡ ( - z ) + ψ ⁡ ( k + 1 ) + ψ ⁡ ( k + m + 1 ) - ψ ⁡ ( a + k + m ) - ψ ⁡ ( c - a - k - m ) ) , scaled-hypergeometric-bold-F 𝑎 𝑎 𝑚 𝑐 𝑧 superscript 𝑧 𝑎 Euler-Gamma 𝑎 𝑚 superscript subscript 𝑘 0 𝑚 1 subscript 𝑎 𝑘 𝑚 𝑘 1 𝑘 Euler-Gamma 𝑐 𝑎 𝑘 superscript 𝑧 𝑘 superscript 𝑧 𝑎 Euler-Gamma 𝑎 superscript subscript 𝑘 0 subscript 𝑎 𝑚 𝑘 𝑘 𝑘 𝑚 Euler-Gamma 𝑐 𝑎 𝑘 𝑚 superscript 1 𝑘 superscript 𝑧 𝑘 𝑚 𝑧 digamma 𝑘 1 digamma 𝑘 𝑚 1 digamma 𝑎 𝑘 𝑚 digamma 𝑐 𝑎 𝑘 𝑚 {\displaystyle{\displaystyle\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(-z)% ^{-a}}{\Gamma\left(a+m\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(m-k-1)!}{k!\Gamma% \left(c-a-k\right)}z^{-k}+\frac{(-z)^{-a}}{\Gamma\left(a\right)}\sum_{k=0}^{% \infty}\frac{(a+m)_{k}}{k!(k+m)!\Gamma\left(c-a-k-m\right)}(-1)^{k}z^{-k-m}\*% \left(\ln\left(-z\right)+\psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left% (a+k+m\right)-\psi\left(c-a-k-m\right)\right),}}
    0 references
    DLMF id
    DLMF:15.8.E8
    0 references
    constraint
    | z | > 1 , | ph ⁡ ( - z ) | < π formulae-sequence 𝑧 1 phase 𝑧 𝜋 {\displaystyle{\displaystyle|z|>1,|\operatorname{ph}\left(-z\right)|<\pi}}
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
     
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