A Poisson-Jacobi-type transformation for the sum \sum_{n=1}^\infty n^{-2m} \exp (-an^2} for positive integer $m$

We obtain an asymptotic expansion for the sum $$S(a;w)=\sum _{n=1}^{\mathrm{\infty }}\frac{e^{-a{n}^2}}{n^w}$$ as $a\to 0$ in $\mathrm{\mid }\arg a\mathrm{\mid }<\pi \mathrm{/}2$ for arbitrary finite $w>0$. The result when $w=2m$, where $m$ is a positive integer, is the analogue of the well-known Poisson-Jacobi transformation for the sum with $m=0$. Numerical results are given to illustrate the accuracy of the expansion.