$$\:\:\:\forall\:\:{x},\:{y}\:\:>\mathrm{0}\:\:\:\:{B}\left({x},{y}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt}\:\:\:\:\:\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\$$$$\left.\mathrm{1}\right)\:{show}\:{that}\:\:\forall\:{x}>\mathrm{0}\:\:\:\:\Gamma\left({x}+\mathrm{1}\right)={x}\Gamma\left({x}\right)\:\:\:\:{and}\:\:{lim}_{{n}−>\infty} \:\frac{{x}\left({x}+\mathrm{1}\right)……\left({x}+{n}\right)}{{n}^{{x}} {n}!}=\frac{\mathrm{1}}{\Gamma\left({x}\right)} \\$$$${and}\:{deduce}\:{that}\:\:{lim}_{{n}−>\infty} \:\frac{\Gamma\left({x}+{n}\right)}{{n}^{{x}} \:\Gamma\left({n}\right)}=\mathrm{1} \\$$$$\left.{b}\right)\:{Prove}\:{that}\:{if}\:\:{a}\:\:{function}\:{f}\:{satisfies}\:\:{f}\left({x}+\mathrm{1}\right)={xf}\left({x}\right)\:\:{et}\:\:\:{lim}_{{n}−>\infty} \:\:\frac{{f}\left({x}+{n}\right)}{{n}^{{x}} \:{f}\left({n}\right)}=\mathrm{1}\:\:{then}\:\forall\:{x}>\mathrm{0}\:\:{f}\left({x}\right)=\:{f}\left(\mathrm{1}\right)\Gamma\left({x}\right) \\$$$$\left.\mathrm{3}\right)\:\:{Show}\:{that}\:\:{B}\left({x}+\mathrm{1},\:{y}\right)=\frac{{x}}{{x}+{y}}{B}\left({x},{y}\right)\:\:\:\:\:\:{B}\left(\mathrm{1},{x}\right)=\frac{\mathrm{1}}{{x}}\: \\$$$$\left.\mathrm{2}\right)\:{Now}\:{let}\:{consider}\:\forall\:{y}\:\:\:{f}\left({x}\right)=\frac{{B}\left({x},{y}\right)\Gamma\left({x}+{y}\right)}{\Gamma\left({y}\right)}\: \\$$$${Show}\:{that}\:\:{f}\:\:{verify}\:\:{the}\:\:{same}\:{property}\:{as}\:\Gamma\:\:\left(\:{just}\:{the}\:{both}\:{proved}\:{up}\:\right)\: \\$$$$\left.\mathrm{3}\right)\:{Deduce}\:\:{that}\:\forall\:{x},{y}>\mathrm{0}\:\:\:{B}\left({x},{y}\right)=\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\$$
$$\left.\mathrm{1}\right)\Gamma\left({x}+\mathrm{1}\right)=\int_{\mathrm{0}} ^{\infty} {t}^{{x}} {e}^{−{t}} {dt}\:\:{by}\:{parts}\:\:{u}\:={t}^{{x}} \:{and}\:{v}^{'} ={e}^{−{t}} \:\Rightarrow \\$$$$\Gamma\left({x}+\mathrm{1}\right)\:=\left[{t}^{{x}} \:{e}^{−{t}} \right]_{\mathrm{0}} ^{\infty} \:−\int_{\mathrm{0}} ^{\infty} {x}\:{t}^{{x}−\mathrm{1}} \:\left(−{e}^{−{t}} \right){dt}\:={x}\int_{\mathrm{0}} ^{\infty} {t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\$$$$={x}\Gamma\left({x}\right) \\$$$${we}\:{have}\:{e}^{{u}} \:={lim}_{{n}\rightarrow+\infty} \left(\mathrm{1}+\frac{{u}}{{n}}\right)^{{n}} \:\Rightarrow{e}^{−{t}} \:={lim}_{{n}\rightarrow+\infty} \left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}} \:\Rightarrow \\$$$$\Gamma\left({x}\right)\:={lim}_{{n}\rightarrow+\infty} \:\:\int_{\mathrm{0}} ^{\infty} \:\:{t}^{{x}−\mathrm{1}} \left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}} \:{dt}={lim}_{{n}\rightarrow+\infty} {W}_{{n}} \\$$$${W}_{{n}} =_{{t}={nu}} \:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\left({nu}\right)^{{x}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{n}} {ndu} \\$$$$={n}^{{x}} \int_{\mathrm{0}} ^{\infty} \:\:{u}^{{x}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{n}} \:{du}\:\:\:\:{integration}\:{by}\:{parts}\:{chow}\:{that} \\$$$$\int_{\mathrm{0}} ^{\infty} {u}^{{x}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{n}} {du}\:=\frac{{n}!}{{x}\left({x}+\mathrm{1}\right)….\left({x}+{n}\right)}\:\Rightarrow \\$$$$\Gamma\left({x}\right)\:={lim}_{{n}\rightarrow+\infty} \:\:\:\frac{{n}^{{x}} {n}!}{{x}\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)….\left({x}+{n}\right)} \\$$