Given the series \begin{align} S_{x}(a) = \sum_{k=1}^{\infty} (-1)^{k+1} \, \binom{x-1}{k} \, L_{k+n-1}(a) \end{align} where $L_{m}(x)$ is the Laguerre polynomial. By using \begin{align} L_{n}(z) = \frac{1}{2 \pi i} \, \int e^{- \frac{z t}{1-t}} \, \frac{dt}{t^{n+1} \, (1-t)} \end{align} it can be determined that \begin{align} S_{x}(a) = L_{n-1}(a) + \frac{(-1)^{x}}{2 \pi i} \, \int e^{- \frac{a t}{1-t}} \, \frac{(1-t)^{x-2}}{t^{n+x-1}} \, dt \end{align}
The question then becomes:
- What is the closed form of the series ?
- How is the remaining contour integral evaluated?
