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I've recently read an article that develop a generalization of the gamma function for negative integers as follows:

$$\Gamma(-x)=\frac{(-1)^x}{xx!}-\frac 1x \Gamma(-x+1)$$ Then setting $\Gamma(0)=-\gamma$ one can calculate $\Gamma(-1)$ and so on

Does this definition comply with the desired properties of the Gamma function? Here you have the link so you can analize better the derivation of the formula http://www.naturalspublishing.com/files/published/t4uv81624e25vq.pdf

EDIT: The article also show this second formula for $\Gamma(-x)$ $$\Gamma(-x)=\frac{(-1)^x}{x!}\phi(x)-\frac{(-1)^x}{x!}\gamma$$ $$=Res(\Gamma,-x)(\phi(x)-\gamma)$$ with $$\phi(x)=\sum_{i=1}^x \frac 1i$$ This formula is then proved by induction on $x$, noticing that for $x=1$ this equation reduces to the first one and then it holds for $x+1$.

EDIT 2: In an attempt of concluding wether or not this definition holds I tried proving that it satisfies the ordinary functional equation for the gamma function, the prove goes as follows: $$-x\Gamma(-x)=\Gamma(-x+1)=\Gamma(-(x-1))$$ $$=-x(\frac{(-1)^x}{x!}\phi(x)-\frac{(-1)^x}{x!}\gamma)$$ $$=-x(\frac{(-1)^x}{x!})(\phi(x)-\gamma)$$ $$=\frac{(-1)^{x+1}}{(x-1)!}(\phi(x)-\gamma)$$ We know that $(-1)^{x+1}=(-1)^{x-1}$ Then $$=\frac{(-1)^{x-1}}{(x-1)!}(\phi(x)-\gamma)$$ But this fails to work, notice that the term $\phi(x)$ remains that way and I can't found any way to modify it to get $\phi(x-1)$. Can someone help me with this?

EDIT 3: Another property I've tried to prove was the Euler's reflection formula. After numerous manipulations I arrived to this equality: $$2x(-1)^x\sum_{n=1}^\infty \frac{(-1)^n}{n^2-x^2} - \frac{(-1)^x}{x} = ln(x) + \sum_{n=2}^\infty \frac{\zeta(n,x+1)}{n}$$ Can someone help me to prove it?

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    One can _define_ something anyway he/she desires. I think either this is valid or invalid is not the right question. The question is, does this _definition_ comply with the desired properties of Gamma function? – 2017-01-21
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    Thanks for your comment, I already changed the question! – 2017-01-21
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    The $\Gamma$ function is already defined on $\mathbb{R}^-$. It is a meromorphic function with simple poles at $-1,-2,-3,\ldots$. – 2017-01-21
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    That's right, but the author tried to assign some kind of principal value for these simple poles. Note that the formula above can be rewritten as $\Gamma(-x)=\frac1x (Res(\Gamma,-x)-\Gamma(-x+1))$. – 2017-01-21
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    Have you checked if this proposed definition satisfies the usual functional equation? [Here](http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html) is an example of an alternative definition. – 2017-01-21
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    That's a good way to start, I'll try to prove the functional equation! And thanks for the article. – 2017-01-21

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