  We consider the asymptotic expansion of the sum \[S_p(a;w)=\sum_{n=1}^\infty
n^{-w}\e^{-an^p}\] as $a\rightarrow 0$ in $|\arg\,a|<\pi/2$ for arbitrary
finite $p>$ and $w>0$. Our attention is concentrated mainly on the case when
$p$ and $w$ are both even integers, where the expansion consists of a {\it
finite} algebraic expansion together with a sequence of increasingly
subdominant exponential expansions. This exponentially small component produces
a transformation for $S_p(a;w)$ analogous to the well-known Poisson-Jacobi
transformation for the sum with $p=2$ and $w=0$. Numerical results are given to
illustrate the accuracy of the expansion obtained.
