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

R B Paris

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arXiv · 1901.00142

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

R B Paris

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Abstract

We examine convergent representations for the sum of a decaying exponential and a Bessel function in the form \[\sum_{n=1}^\infty \frac{e^{-an}}{(\frac{1}{2} bn)^\nu}\,J_\nu(bn),\] where $J_\nu(x)$ is the Bessel function of the first kind of order $\nu>-1/2$ 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 $a\to 0$ with $b<2\pi$ fixed. A similar result is derived for the sum when the Bessel function is replaced by the modified Bessel function $K_\nu(x)$. The alternating versions of these sums are also mentioned.

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R B Paris. 2019-01-01. An expansion for the sum of a product of an exponential and a Bessel function. https://arxiv.org/abs/1901.00142

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