A summation of a series with power, exponential and factorial $\sum_{k=0}^\infty {\frac{k^{C_1} {C_2}^k}{k!}}$ 
where $C_1$ is a positive integer and $C_2$ is a real number. Is there a close form or an approximation of the result when the summation converges? 

Note that we can write
$$ x^{C_1} = \sum_{l=0}^{C_1} a_l x (x-1) \cdots (x-l+1) $$
for some coefficients $(a_l)$. Then by noting that
$$ \sum_{k=0}^{\infty} \frac{k(k-1)\cdots(k-l+1)}{k!} C_2^{k} = C_2^{l} \left. \frac{d^l e^{x}}{dx^l} \right|_{x=C_2} = C_2^{l}e^{C_2}, $$
we obtain
$$ \sum_{k=0}^{\infty} \frac{k^{C_1} C_2^{k}}{k!} = e^{C_2} \sum_{l=0}^{C_1} a_l C_2^{l}. $$
For example, if $C_1 = 10$ then
\begin{align*}
a_0 &= 0, & a_1 &= 1, & a_2 &= 511, & a_3 &= 9330, \\
a_4 &= 34105, & a_5 &= 42525, & a_6 &= 22827, & a_7 &= 5880, \\
a_8 &= 750, & a_9 &= 45, & a_{10} &= 1.
\end{align*}
and we have
\begin{align*}
\sum_{k=0}^{\infty} \frac{k^{10} C_2^{k}}{k!}
 = e^{C_2} \sum_{l=0}^{C_1} a_l C_2^{l}
&= e^{C_2} \big( C_2^{10}+45 C_2^9+750 C_2^8+5880 C_2^7+22827 C_2^6 \\
&\qquad +42525 C_2^5+34105 C_2^4+9330 C_2^3+511 C_2^2+C_2 \big).
\end{align*}
