\begin{equation}\sum_{m=1}^{\infty}\frac{a_1^3 S_m^{(3)} (u-1)^{m-1} \left(\frac{x}{x-1}\right)^m}{m!}\end{equation} Does somebody know the result of this resummation?
Note: $S_m^{(3)} $ belongs to the Stirling number of the first kind https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind If you see the wiki page "Table of values" $S_m^{(3)} $ corresponds to the third column.
Please let me know if addition information is needed.