A Poisson-Jacobi-type transformation for the sum ∑_n=1^∞ n^-2m exp (-an² for positive integer m

2015/01/04 by R. B. Paris, Paris, R. B.
Mathematics · #30E15 #33B10 #34E05 #41A30 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #math.CA #msc:30E15 #msc:33B10 #msc:34E05 #msc:41A30

paper · pdf · doi:10.48550/arxiv.1501.00685

7 pages, 0 figures

arxiv created 2015/01/04 · openalex publication_date 2015/01/04 · arxiv updated 2015/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain an asymptotic expansion for the sum S(a;w)=∑_n=1^∞ \frace^-an2n^w as a→ 0 in |arg a|<π/2 for arbitrary finite w>0. The result when w=2m, where m is a positive integer, is the analogue of the well-known Poisson-Jacobi transformation for the sum with m=0. Numerical results are given to illustrate the accuracy of the expansion.

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