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

TL;DR

This paper derives an asymptotic expansion for a sum of modified Bessel functions with quadratic argument, analyzing the Stokes phenomenon and providing numerical validation as the parameter approaches zero.

Contribution

It presents a detailed asymptotic expansion for the sum involving modified Bessel functions, including the behavior on the Stokes line as the parameter tends to zero.

Findings

Identification of the Stokes line at the positive real axis.

Description of the smooth transition of exponentially small terms across the Stokes line.

Numerical results confirming the asymptotic expansion's accuracy.

Abstract

We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by \[S_\nu(a)=\sum_{n\geq 1} (\frac{1}{2} an^2)^{-\nu} K_\nu(an^2)\qquad (|\arg\,a|<\pi/2)\] 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 $a\to 0$ through positive values. Numerical results are presented to support the asymptotic theory.