2019/04/30 by R. B. Paris, R B Paris, Paris, R B
Mathematics · #33C10 #34E05 41A30 #41A60 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #math.CA #msc:33C10 #msc:34E05 #msc:41A30 #msc:41A60
paper · pdf · doi:10.48550/arxiv.1905.00009
13 pages, 0 figures. arXiv admin note: text overlap with arXiv:1812.10764
arxiv created 2019/04/30 · openalex publication_date 2019/04/30 · arxiv updated 2019/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We examine the sum of modified Bessel functions with argument depending non-linearly on the summation index given by Sν,p(a)=∑n≥ 1 (anp/2)-ν Kν(anp) (a>0, 0≤ν<1) as the parameter a→ 0+, where p denotes an integer satisfying p≥ 2. This extends previous work for the cases p=1 (linear) and p=2 (quadratic). The expansion as a→0+ 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.