Γ(s+1)=sΓ(s)

\(\Gamma(s+1)=s\Gamma(s)\)

$$ \begin{eqnarray} \Gamma(s+1)&=&\int_{0}^{\infty}e^{-t}t^{s\color{red}{+1-1}}\mathrm{d}t \;\cdots\;\Gamma(s)=\int_{0}^{\infty}e^{-t}t^{s-1}\mathrm{d}t \\&=&\int_{0}^{\infty}e^{-t}t^{s}\mathrm{d}t \\&=&\int_{0}^{\infty}\left\{-e^{-t}\right\}^\prime t^{s}\mathrm{d}t \;\cdots\;\frac{\mathrm{d}}{\mathrm{d}t}\left(-e^{-t}\right)=\left\{-e^{-t}\right\}^\prime=e^{-t} \\&=&\left[-e^{-t} t^{s}\right]_0^\infty-\int_{0}^{\infty}\left\{-e^{-t}\right\}\left\{ st^{s-1} \right\}\mathrm{d}t \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/02/blog-post_7.html}{\int f^\prime g\;\mathrm{d}x= fg-\int f g^\prime\;\mathrm{d}x} \\&=&\left[\left(\lim_{t\rightarrow \infty} -\frac{t^{s}}{e^{t}}\right)-\left(-\frac{0^{s}}{e^{0}}\right)\right]+s\int_{0}^{\infty}e^{-t}t^{s-1}\mathrm{d}t \;\cdots\;\int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x \\&=&\left[0-0\right]+s\int_{0}^{\infty}e^{-t}t^{s-1}\mathrm{d}t \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/xnex.html}{\lim_{t\rightarrow \infty} \frac{t^{s}}{e^{t}}=0} \\&=&s\Gamma(s) \;\cdots\;\Gamma(s)=\int_{0}^{\infty}e^{-t}t^{s-1}\mathrm{d}t \end{eqnarray} $$

\(\Gamma(s+2)=(s+1)s\Gamma(s)\)

$$ \begin{eqnarray} \Gamma(s+2)&=&(s+1)\Gamma(s+1) \\&=&(s+1)s\Gamma(s) \end{eqnarray} $$