On Infinite Series of Bessel functions of the First Kind: $\sum_ν J_{Nν+p}(x), \sum_ν(-1)^νJ_{Nν+p}(x)$ (2211.01148v1)

Abstract: Infinite series of Bessel function of the first kind, $\sum_\nu^{\pm\infty} J_{N\nu+p}(x)$, $\sum_\nu^{\pm\infty} (-1)^\nu J_{N\nu+p}(x)$, are summed in closed form. These expressions are evaluated by engineering a Dirac comb that selects specific sequences within the Bessel series.