Can this generalization of the exponential function be expressed using more familiar functions?

Question

For each $n \in \mathbb{R}_{\neq -1}$, define:

$$\exp_n(x) = \sum_{i=0}^\infty \frac{x^i}{\prod_{j=1}^i(j+n)} = 1 + \frac{x}{1+n}+\frac{x^2}{(1+n)(2+n)}+\cdots$$

Note that the $n=0$ case corresponds to the familiar exponential function.

Question. Can this generalization of the exponential function be expressed using more familiar functions?

I'm mainly interested in the case $n \in \{0,1,2,3,\ldots\}$

Answer 1

Using Pochhamer symbols $$\prod_{j=1}^i(j+n)=(n+1)_i$$ which make $$\exp_n(x)=\frac{e^x}{ x^{n}}\, \left(\Gamma (n+1)-n\, \Gamma (n,x)\right)$$ where appear the complete and incomplete gamma functions.