  We consider the asymptotic expansion of the functional series
\[S_{\mu}^\pm(a;\lambda)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-\lambda
n}}{(n^2+a^2)^\mu}\] for $\lambda>0$ and $\mu\geq0$ as $|a|\to \infty$ in the
sector $|\arg\,a|<\pi/2$. The approach employed consists of expressing
$S_{\mu}^\pm(a;\lambda)$ as a contour integral combined with suitable
deformation of the integration path. Numerical examples are provided to
illustrate the accuracy of the various expansions obtained.
