Relation between two gamma functions

Question

Does anyone know the relation between these two gamma functions?

1st) Gamma[1 + c, a (1 + b)]

2nd) Gamma[c, a (1 + b)]

The question is: may I write the 1st like the 2nd times something?

thank you very much

Answer 1

Let's note $x:=a(1+b)$ then for positive $c$ and $x$ we have : $$\Gamma(1 + c,\; x)-c\;\Gamma( c,\, x)=x^c\;e^{-x}$$

To show this use the definition and apply integration by parts :

\begin{align} \Gamma(1 + c,\; x)&=\int_x^\infty t^c\,e^{-t}\;dt\\ &=\left. -t^c\,e^{-t}\right|_x^\infty+c\,\int_x^\infty t^{c-1}\,e^{-t}\;dt\\ &=x^c\,e^{-x}+c\;\Gamma( c,\, x)\\ \end{align}