An asymptotic expansion for a sum of modified Bessel functions with quadratic argument

We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by [S_ u(a)=sum_{ngeq 1} (frac{1}{2} an^2)^{- u} K_ u(an^2)qquad (|arg,a|<pi/2)] as the parameter $|a|to 0$. It is shown that the positive real $a$-axis is a Stokes line, where an infinite number of increasingly subdominant exponentially small terms present in the asymptotic expansion undergo a smooth, but rapid, transition as this ray is crossed. Particular attention is devoted to the details of the expansion on the Stokes line as $ato 0$ through positive values. Numerical results are presented to support the asymptotic theory.