The Bessel function of the first kind JN(kx) is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind IN(kx). The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for JN(kx) useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving 1F2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1/ (2i3j5k7l11m13n17o19p) multiplying powers of the coefficient k.