How to find $\text{summation formula}$ for the following power series?
$$ \sum_{n=0}^{\infty} \frac{((an+b)!)^{an+b}}{r+((an+b)!)^{an+b}} p_k(n) \cdot \frac{x^n}{(an+b)!} , \ r \in \mathbb{Q}^{+}, \ a \in \mathbb{N}, \ b \in \mathbb{N} \cup \{0 \},$$ where $p_k(n)$ is some polynomial in $n$ of degree $k$.
The above power series converges everywhere in $\mathbb{R}$.
I need to find general summation formula. But, how to find this?
My purpose is to show when the sum becomes a rational number by finding the summation formula.
One thing seems to me that using some recurrence relation we can drive the summation formula.
Can someone give me some hints to derive the summation formula?