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

2018/12/27 by R. B. Paris, Paris, R. B. · 1 citation
Mathematics · #33C10 #3405 #41A30 #41A60 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:33C10 #msc:3405 #msc:41A30 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1812.10764

13 pages, 0 figures

openalex publication_date 2018/12/27 · openalex created_date 2019/01/11 · arxiv created 2019/03/06 · arxiv updated 2019/03/07 · openalex updated_date 2026/07/28

Abstract

We examine the sum of modified Bessel functions with argument depending quadratically on the summation index given by S_ν(a)=∑_{n≥ 1} ((1)/(2) an²)^-ν K_ν(an²) (|arg a|<π/2) as the parameter |a|→ 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→ 0 through positive values. Numerical results are presented to support the asymptotic theory.

Citations

• The asymptotic expansion of a generalisation of the Euler-Jacobi series

Cited by

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

Related

• Asymptotics of a sum of modified Bessel functions with non-linear argument
• Asymptotic expansion of Mathieu-Bessel series. II
• An expansion for the sum of a product of an exponential and a Bessel function. II
• An expansion for the sum of a product of an exponential and a Bessel function
• Asymptotic expoansions of mathieu-Bessel series. I