Asymptotics of a Mathieu-Gaussian series

We consider the asymptotic expansion of the functional series $$S_{\mu ,\gamma }(a;\lambda )=\sum _{n=1}^{\mathrm{\infty }}\frac{n^{\gamma }{e}^{-\lambda n^2\mathrm{/}{a}^2}}{(n^2+a^2{)}^{\mu }}$$ for real values of the parameters $\gamma$, $\lambda >0$ and $\mu \ge 0$ as $\mathrm{\mid }a\mathrm{\mid }\to \mathrm{\infty }$ in the sector $\mathrm{\mid }\arg a\mathrm{\mid }<\pi \mathrm{/}4$. For general values of $\gamma$ the expansion is of algebraic type with terms involving the Riemann zeta function and a terminating confluent hypergeometric function. Of principal interest in this study is the case corresponding to even integer values of $\gamma$, where the algebraic-type expansion consists of a finite number of terms together with a contribution comprising an infinite sequence of increasingly subdominant exponentially small expansions. This situation is analogous to the well-known Poisson-Jacobi formula corresponding to the case $\mu =\gamma =0$. Numerical examples are provided to illustrate the accuracy of these expansions.