$$\begin{align*}
\mathrm{exps}^{(m)}_s(x)
=&\frac{1}{s}\sum_{k=0}^\infty{s-1}\left[\left(e^{\frac{m}{s}\cdot2\pi i}\right)^ke^{\left(e^{\frac{2\pi i}{s}}\right)^kx}\right]\\=&\frac{1}{s}\sum_{k=0}^\infty{s-1}\left[\exp\left(\frac{2km\pi}{s}i+\exp\left(\frac{2k\pi}{s}i\right)x\right)\right]
\end{align*}$$