I'm trying to prove the following equality: $$ \gamma(r,x)=\int_0^x \Gamma(r,y) \sum_{m=1}^\infty\frac{e^{-(x-y) } (x-y) ^{mr-1}}{\Gamma(mr)}dy, $$ where $\Gamma(\cdot)$ is the Gamma function, $\gamma(\cdot, \cdot)$ the lower incomplete Gamma function and $\Gamma(\cdot, \cdot)$ the upper incomplete Gamma function: $$ \Gamma(r) = \int_0^{+\infty} t^{r-1}e^{-t}\ dt, \quad \Gamma(r,x) = \int_x^{+\infty} t^{r-1}e^{-t}\ dt, \quad \gamma(r,x) = \int_0^{x} t^{r-1}e^{-t}\ dt, $$ and $x>0, r>0.$
I tried to use Mathematica and it was not successful. I coded it numerically and did not find any counter example, so it seems to be true. Help would be gratefully acknowledged!