  We examine the sum of modified Bessel functions with argument depending
non-linearly on the summation index given by \[S_{\nu,p}(a)=\sum_{n\geq 1}
(an^p/2)^{-\nu} K_\nu(an^p)\qquad (a>0,\ 0\leq\nu<1)\] as the parameter $a\to
0+$, where $p$ denotes an integer satisfying $p\geq 2$. This extends previous
work for the cases $p=1$ (linear) and $p=2$ (quadratic). The expansion as
$a\to0+$ consists of an infinite number of asymptotic sums involving the
Riemann zeta function, which when optimally truncated lead to remainder terms
that are exponentially small in the parameter $a$. The number of these
exponentially small terms associated with each optimally truncated asymptotic
sum is found to increase with $p$.
