In deriving the solution to a differential equation I arrived at the following series expression:
$f(x;a)=\sum_{n=0}^{\infty}\cfrac{n!(1-x)^n}{(1-a)(2-a)\cdots(n-a)};\quad 0\le x \le 1$
where $a$ is a very small parameter of the equation, and is NOT an integer (otherwise the formula is not valid). After some brief analysis I found this series to be convergent, but I couldn't find a closed form expression.
Does it belong to any existing families of special functions?
Indeed it is - it's a kind of hypergeometric function. More specifically, we rewrite it through a series of steps as
$$\begin{align} f(x; a) &= \sum_{n=0}^{\infty} \frac{n!}{(1 - a)(2 - a) \cdots (n - a)} \frac{n!}{n!} (1 - x)^n\\ &= \sum_{n=0}^{\infty} \frac{n! n!}{(1 - a)(2 - a) \cdots (n - a)} \frac{(1 - x)^n}{n!}\\ &= \sum_{n=0}^{\infty} \frac{n! n!}{(1 - a)([1 - a] + 1) \cdots ([1 - a] + n - 1)} \frac{(1 - x)^n}{n!}\end{align}$$
and now converting to rising factorials, using $n! = 1^{(n)}$, we get
$$f(x; a) = \sum_{n=0}^{\infty} \frac{1^{(n)}\ 1^{(n)}}{[1 - a]^{(n)}} \frac{(1 - x)^n}{n!}$$
which we can read off now as
$$f(x; a) =\ _2 F_1\left(\begin{matrix}1, 1 \\ 1 - a\end{matrix} \Big| 1 - x\right)$$