Some Neumann-Bessel series and the Laplacian on polygons

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].