One of the ways to write the gamma function is:
$$\Gamma(z)=\int_0^\infty e^{-t} t^{z-1}\,dt, \quad \operatorname{Real}(z)>0$$
Another one to write it is:
$$\Gamma(z)=2\int_0^\infty e^{-t^2}t^{2z-1}\,dt \quad \operatorname{Real}(z)>0$$
Can I generalize it? writing it as:
$$\Gamma(z)=n\int_0^\infty e^{-t^n}t^{nz-1} \, dt, \quad \operatorname{Real}(z)>0$$