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 $S(a;w)=\sum_{n=1}^\infty \frac{e^{-an^2}}{n^{w}}$ as $a\rightarrow 0$ in $|\arg\,a|<π/2$ 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.

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