Asymptotic expoansions of mathieu-Bessel series. I

2019/07/03 by R. B. Paris, Paris, R B
Mathematics · #30E15 #30E20 #34E05 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1907.01812

openalex publication_date 2019/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the asymptotic expansion of the Mathieu-Bessel series S_ν(a,b)=∑_n=1^∞ (n^γJ_ν(nb/a))/((n²+a²)^μ), (μ, bgt;0, γ, ν∈ \bf R) as a→+∞ with the other parameters held fixed, where J_ν(x) is the Bessel function of the first kind of order ν. A special case arises when γ+ν is a positive even integer, where the expansion comprises finite algebraic terms together with an exponentially small expansion. Numerical examples are presented to illustrate the accuracy of the various expansions. The expansion of the alternating variant of S_ν(a,b) is considered. The series when the J_ν(x) function is replaced by the Bessel function Y_ν(x) is briefly mentioned.

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