DLMF:16.8.E9

MaRDI QIDQ5925

• Label: DLMF:16.8.E9

Digital Library of Mathematical Functions ID 16.8.E9

${\displaystyle{\displaystyle\left({\textstyle\ifrac{\prod\limits_{k=1}^{q+1}\Gamma\left(a_{k}\right)}{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{q+1}F_{q}}\left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=\sum_{j=1}^{q+1}\left(z_{0}-z\right)^{-a_{j}}\sum_{n=0}^{\infty}\frac{\Gamma\left(a_{j}+n\right)}{n!}\*\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\Gamma\left(a_{k}-a_{j}-n\right)}{\prod\limits_{k=1}^{q}\Gamma\left(b_{k}-a_{j}-n\right)}}\right)\*{{}_{q+1}F_{q}}\left({a_{1}-a_{j}-n,\dots,a_{q+1}-a_{j}-n\atop b_{1}-a_{j}-n,\dots,b_{q}-a_{j}-n};z_{0}\right)\left(z-z_{0}\right)^{-n}.}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}$: alternatively F q p ( a ; b ; z ) or F q p ( a b ; z ) generalized hypergeometric function
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle q}}$: nonnegative integer
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}$: real or complex parameters
• ${\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}$: real or complex parameters
 

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