Γ(s+1)=sΓ(s)

$\mathrm{\Gamma }(s+1)=s\mathrm{\Gamma }(s)$

$$\begin{array}{rcl}\mathrm{\Gamma }(s+1)& =& {\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s+1-1}\mathrm{d}t\cdots \mathrm{\Gamma }(s)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^s\mathrm{d}t\\ & =& {\int }_0^{\mathrm{\infty }}{\{-e^{-t}\}}^{\prime }{t}^s\mathrm{d}t\cdots \frac{\mathrm{d}}{\mathrm{d}t}(-e^{-t})={\{-e^{-t}\}}^{\prime }=e^{-t}\\ & =& {[-e^{-t}{t}^s]}_0^{\mathrm{\infty }}-{\int }_0^{\mathrm{\infty }}\{-e^{-t}\}\left\{s{t}^{s-1}\right\}\mathrm{d}t\cdots \int f^{\prime }g\mathrm{d}x=fg-\int f{g}^{\prime }\mathrm{d}x\\ & =& [(\underset{t\to \mathrm{\infty }}{lim}-\frac{t^s}{e^t})-(-\frac{0^s}{e^0})]+s{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}\mathrm{d}t\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& [0-0]+s{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}\mathrm{d}t\cdots \underset{t\to \mathrm{\infty }}{lim}\frac{t^s}{e^t}=0\\ & =& s\mathrm{\Gamma }(s)\cdots \mathrm{\Gamma }(s)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}\mathrm{d}t\end{array}$$

$\mathrm{\Gamma }(s+2)=(s+1)s\mathrm{\Gamma }(s)$

$$\begin{array}{rcl}\mathrm{\Gamma }(s+2)& =& (s+1)\mathrm{\Gamma }(s+1)\\ & =& (s+1)s\mathrm{\Gamma }(s)\end{array}$$