expectation of incomplete gamma

Question

Is the expectation of the (upper/lower) incomplete gamma function known?

$$\int_0^{+\infty} x \Gamma(A, x) \mathrm dx$$

Answer 1

Using formula 8.14.4 in the DLMF (the Mellin transform of the upper incomplete gamma function $\Gamma(b,x)$) and specializing that formula to the case $a=2$ gives the result

$$\int_0^{+\infty} x \Gamma(A, x) \mathrm dx=\frac{\Gamma(A+2)}{2}$$

which is valid for $A > -2$. Alternatively, using the regularized form $Q(b,x)=\frac{\Gamma(b,x)}{\Gamma(b)}$, we have the result

$$\int_0^{+\infty} x Q(A, x) \mathrm dx=\frac{A(A+1)}{2}$$

Answer 2

It's not an expectation as stated because the supposed pdf does not integrated to unity. If $A+2>0$, the integral is $\Gamma(A+2)/2$