Exponentially small expansions associated with a generalised Mathieu series

We consider the generalised Mathieu series $$\sum_{n=1}^\infty \frac{n^\gamma}{(n^\lambda+a^\lambda)^\mu}\qquad (\mu>0)$$ when the parameters $\lambda$ ($>0$) and $\gamma$ are even integers for large complex $a$ in the sector $|\arg\,a|<\pi/\lambda$. The asymptotics in this case consist of a {\it finite} algebraic expansion together with an infinite sequence of increasingly subdominant exponentially small expansions. When $\mu$ is also a positive integer it is possible to give closed-form evaluations of this series. Numerical results are given to illustrate the accuracy of the expansion obtained.