An expansion for the sum of a product of an exponential and a Bessel function

2019/01/01 by Paris, R B
#33C05 #33C10 #33C20 #41A30 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1901.00142

Abstract

We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form ∑_n=1^∞ \frace^-an((1)/(2) bn)^ν J_ν(bn), where J_ν(x) is the Bessel function of the first kind of order ν>-1/2 and a, b are positive parameters. By means of a double Mellin-Barnes integral representation we obtain a convergent asymptotic expansion that enables the evaluation of this sum in the limit a→ 0 with b<2π fixed. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function K_ν(x). The alternating versions of these sums are also mentioned.

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