DLMF:16.8.E9 (Q5228)

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DLMF:16.8.E9
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    Statements

    Defining formula
    ( ∏ k = 1 q + 1 Γ ⁡ ( a k ) / ∏ k = 1 q Γ ⁡ ( b k ) ) ⁢ F q q + 1 ⁡ ( a 1 , … , a q + 1 b 1 , … , b q ; z ) = ∑ j = 1 q + 1 ( z 0 - z ) - a j ⁢ ∑ n = 0 ∞ Γ ⁡ ( a j + n ) n ! ⁢ ( ∏ k = 1 k ≠ j q + 1 Γ ⁡ ( a k - a j - n ) / ∏ k = 1 q Γ ⁡ ( b k - a j - n ) ) ⁢ F q q + 1 ⁡ ( a 1 - a j - n , … , a q + 1 - a j - n b 1 - a j - n , … , b q - a j - n ; z 0 ) ⁢ ( z - z 0 ) - n . superscript subscript product 𝑘 1 𝑞 1 Euler-Gamma subscript 𝑎 𝑘 superscript subscript product 𝑘 1 𝑞 Euler-Gamma subscript 𝑏 𝑘 Gauss-hypergeometric-pFq 𝑞 1 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑞 1 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 superscript subscript 𝑗 1 𝑞 1 superscript subscript 𝑧 0 𝑧 subscript 𝑎 𝑗 superscript subscript 𝑛 0 Euler-Gamma subscript 𝑎 𝑗 𝑛 𝑛 superscript subscript product 𝑘 1 𝑘 𝑗 𝑞 1 Euler-Gamma subscript 𝑎 𝑘 subscript 𝑎 𝑗 𝑛 superscript subscript product 𝑘 1 𝑞 Euler-Gamma subscript 𝑏 𝑘 subscript 𝑎 𝑗 𝑛 Gauss-hypergeometric-pFq 𝑞 1 𝑞 subscript 𝑎 1 subscript 𝑎 𝑗 𝑛 … subscript 𝑎 𝑞 1 subscript 𝑎 𝑗 𝑛 subscript 𝑏 1 subscript 𝑎 𝑗 𝑛 … subscript 𝑏 𝑞 subscript 𝑎 𝑗 𝑛 subscript 𝑧 0 superscript 𝑧 subscript 𝑧 0 𝑛 {\displaystyle{\displaystyle\left({\textstyle\ifrac{\prod\limits_{k=1}^{q+1}% \Gamma\left(a_{k}\right)}{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}% \right){{}_{q+1}F_{q}}\left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z% \right)=\sum_{j=1}^{q+1}\left(z_{0}-z\right)^{-a_{j}}\sum_{n=0}^{\infty}\frac{% \Gamma\left(a_{j}+n\right)}{n!}\*\left({\textstyle\ifrac{\prod\limits_{% \begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\Gamma\left(a_{k}-a_{j}-n\right)}{\prod\limits_{k=% 1}^{q}\Gamma\left(b_{k}-a_{j}-n\right)}}\right)\*{{}_{q+1}F_{q}}\left({a_{1}-a% _{j}-n,\dots,a_{q+1}-a_{j}-n\atop b_{1}-a_{j}-n,\dots,b_{q}-a_{j}-n};z_{0}% \right)\left(z-z_{0}\right)^{-n}.}}
    0 references
    DLMF id
    DLMF:16.8.E9
    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.m2aadec
    0 references
    Q11125
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1acdec
    0 references
     
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