  We obtain an asymptotic expansion for the sum \[S(a;w)=\sum_{n=1}^\infty
\frac{e^{-an^2}}{n^{w}}\] as $a\rightarrow 0$ in $|\arg\,a|<\pi/2$ 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.
