Asymptotics of a sum of modified Bessel functions with non-linear argument

TL;DR

This paper derives the asymptotic behavior of a sum involving modified Bessel functions with non-linear arguments as a parameter approaches zero, extending previous linear and quadratic cases and revealing exponentially small remainder terms.

Contribution

It generalizes the asymptotic analysis of sums of modified Bessel functions to non-linear arguments for any integer p ≥ 2, revealing new exponential decay terms.

Findings

Asymptotic expansion involves infinite sums with Riemann zeta functions.

Optimal truncation yields exponentially small remainder terms.

Number of exponential terms increases with p.

Abstract

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 $p$.