Characterizing $\frac{m^k - e^{-x}(m+1)^k}{k!}$

Question

I'm trying to characterize the behavior this function: $l_k(m,x)=\frac{m^k - e^{-x}(m+1)^k}{k!}$.

I was wondering whether either of these functions are well-known in the probability theory and statistics literature:

$l_k(m,x)$

or

$\sum_{k=0}^m x^k l_k(m,x)$.

I appreciate if some similar papers in which these types of functions are used can be mentioned.

Answer 1

I suppose, since $m$ is a constant, that the second question is $$S_m=\sum_{k=0}^m x^\color{red}{k} \,l_k(m,x)$$ I am curious to know in which particular context you had to work it; in my research group, we had the same problem in thermodynamics and the solution is $$S_m=e^{mx} \,\frac{\Gamma (m+1,m x)-\Gamma (m+1,(m+1) x)}{\Gamma (m+1)}$$

Similarly $$T_m=\sum_{k=0}^m \,l_k(m,x)=e^m \,\frac{\Gamma (m+1,m)-e^{1-x} \Gamma (m+1,m+1)}{\Gamma (m+1)}$$