Do you recognize this infinite series? $\sum_{n=0}^\infty \frac 1{(1+an)^c} \frac{b^n}{n!} $

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I came by this infinite series

$$\sum_{n=0}^\infty \frac 1{(1+an)^c} \frac{b^n}{n!} $$

Is there some special function that can have this form?

$c$ can be assumed to be a positive integer. While $a,b$ are positive real numbers