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Some Results on the Gamma Function for Negative Integers |
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PP: 173-176 |
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Author(s) |
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Brian Fisher,
Adem Kılıcman,
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Abstract |
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| The Gamma function $\Gamma^{(s)} (-r)$ is defined by \beqa \Gamma^{(s)}(-r)= \Nlim_{\epsilon\to 0}\int _\epsilon^ \infty t^{-r-1} \ln^s t\,e^{-t}\,dt \eeqa for $r,s=0,1,2,\ldots ,$ where $N$ is the neutrix having domain $N=\{ \epsilon :\ 0<\epsilon< \infty\}$ with negligible functions finite linear sums of the functions $$\epsilon^ \lam \ln ^{s-1} \epsilon, \ \ \ln^s\epsilon:\ \ \lam<0,\ \ s=1,2,\ldots$$ and all functions which converge to zero in the normal sense as $\epsilon$ tends to zero.\ \noindent In the classical sense $Gamma$ functions is not defined for the negative integer. In this study, it is proved that \beqa \Gamma (-r) ={(-1)^r\over r!}\phi(r)- {(-1)^r \over r!}\, \gamma\eeqa for $r=1,2,\ldots,$ where $$\phi(r) =\sum _{i=1} ^r {1\over i}.$$ \noindent Further results are also proved. |
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