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\end{array}  \end{equation}  Above we have used that $(z-1)\Gamma(z)-\Gamma(z+1)=-\Gamma(z)$ $(z-1)\Gamma(z)-\Gamma(z+1)=-\Gamma(z)$.  This derivative has explicity operated on functions of $x$, which makes it still in some sense "with respect to x". One could imagine multiple operations on multidimensional functions...  Contradictory examples to this are