A binomial multiplied by a shifted poisson

Question

I am trying to simplify this expression $$ \sum_{k\geq 0} \frac{(k+x)!}{k!} \frac{b^{s+k+x}}{(s+k+x)!} $$ to an expression with finite summation. I am able to do for x = 0 and 1, but not able to reduced for x>1. If anyone has similar experience in solving this or is able to reduced, can you please help to share.

Answer 1

Assuming that $s$ is the shift parameter and is thus always a non-negative integer, you can always express this sum in terns of powers and exponentials of $b$ and incomplete gamma functions.

For example, when $x=2$ the expression is $$ \frac{b^{s+2}}{s+1)!} \left[1+b-s+\frac{e^b}{b^{s+2}}\left(b^2 -2bs+s(s+1)\right) \left(\Gamma(s+2)-\gamma(s+2,b)\right)\right] $$

Since even for $x=0$ your expression involves an incomplete gamma function, this is as good as you are going to get.