The upper incomplete gamma function is defined as
$$\Gamma\left(s,x\right)=\int_{x}^{\infty}t^{s-1}e^{-t}dt$$
Can this expression be normalized to obtain a distribution on $x$? i.e. does the integral
$$\int_{0}^{\infty}\Gamma\left(s,x\right)dx=\int_{0}^{\infty}\int_{x}^{\infty}t^{s-1}e^{-t}dtdx$$
converge? If so, to what? (i.e. what is the normalizing constant). Is this a special case of a known distribution?
One can write the tail of a Gamma$(s,1)$-distributed variable $X$ in terms of the upper incomplete Gamma function $\Gamma(s,x)$: $$\Bbb P(X>x)=\frac1{\Gamma(s)}\int_x^\infty t^{s-1}\,\mathrm e^{-t}\,\mathrm dt=\frac{\Gamma(s,x)}{\Gamma(s)}.$$ Therefore $$\int_0^\infty\Gamma(s,x)\,\mathrm dx=\Gamma(s)\int_0^\infty\Bbb P(X>x)\,\mathrm dx=\Gamma(s)\,\Bbb E[X]=s\Gamma(s)=\Gamma(s+1).$$