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?

  • $\begingroup$ You are missing a bracket in the denominator, can you add one so that the summation makes sense? $\endgroup$ – Peter Foreman Mar 6 at 7:47
  • $\begingroup$ @PeterForeman, I have edited my question $\endgroup$ – M. A. SARKAR Mar 6 at 7:59
  • $\begingroup$ Why do you think there should be a closed form for this series? $\endgroup$ – Greg Martin Mar 6 at 8:36
  • $\begingroup$ @GregMartin, I am not asking the exact sum, but I need the summation formula based on the polynomial $p_k(n)$ for different particular cases. I need just the summation formula to show that the sum of the series can be a rational number. Can you give some hintz ? $\endgroup$ – M. A. SARKAR Mar 6 at 8:41
  • $\begingroup$ What do you mean by "the summation formula", if not a closed form? $\endgroup$ – Greg Martin Mar 6 at 20:30

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