Is there any closed form for this series?

Question

It's a power series that I found during the computation for my research.

\begin{equation*} \sum_{k=0}^n \binom{n}{k}\frac{n!}{(n-k)!}x^{n-k}(-1)^k. \end{equation*}

Without the annoying term of $\frac{n!}{(n-k)!}$, it is clearly simplified to $(x-1)^n$ due to binomial theorem.

Isn't there any name for this series? Can I simplify it as a closed-form polynomial?

Answer 1

You can write it this way: $$\sum_{k=0}^n \binom{n}{k}\frac{n!}{(n-k)!}x^{n-k}(-1)^k = \sum_{k=0}^n \binom{n}{k}(-1)^k\frac{d^k}{dx^k} (x^n) = \left(I-\frac d{dx}\right)^n x^n $$

Answer 2

It can be rewritten as $\displaystyle\sum_{k=0}^n{n\choose k}^2k!x^{n-k}(-1)^k$, and, unless you're willing to accept hypergeometric functions as closed form, the answer is no. My advice would be for you to try and approximate it asymptotically.