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Above we have used (or discovered depending on your point of view!) that $(z-1)\Gamma(z)-\Gamma(z+1)=-\Gamma(z)$. I.e $(n-1)(n-1)!-n!=-(n-1)!$, which allows us to discover the recursive formula $n!=n(n-1)!$. Of course, it also applies to the complex functionality etc.  Then reused this again an used/discovered that $(1-2z)\Gamma(z+1)+\Gamma(z+2)+(z-1)^2\Gamma(z)=\Gamma(z)$...  This might not be so obvious and upon rearranging elucidates that \begin{equation}  n!=(2n-3)(n-1)!+(4n-3-n^2)(n-2)!  \end{equation}  Altohugh if one knows the inner workings of the gamma function this is perhaps obvious, we could potentially using this method, discover identities for horrendously complicated functions without knowing what they were... I will demonstrate below later on.