Journal of Mathematical Physics, Volume 62, Issue 5, May 2021. Several sums of Neumann series with Bessel and trigonometric functions are evaluated, as finite sums of trigonometric functions. They arise from a generalization of the Neumann expansion of the eigenstates of the Laplacian in regular polygons. A simple accurate approximation of J0(x) is found on the interval [0, 2].

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