  The Bessel function of the first kind $J_{N}\left(kx\right)$ is expanded in a
Fourier-Legendre series, as is the modified Bessel functions of the first kind
$I_{N}\left(kx\right)$. The purpose of these expansions in Legendre polynomials
was not an attempt to rival established \emph{numerical methods} for
calculating Bessel functions, but to provide a form for $J_{N}\left(kx\right)$
useful for \emph{analytical} work in the area of strong laser fields, where
analytical integration over scattering angles is essential. Despite their
primary purpose, we can easily truncate the series at 21 terms to provide
33-digit accuracy that matches IEEE extended precision in some compilers. The
analytical theme is furthered by showing that infinite series of like-powered
contributors (involving $\,_{2}F_{3}$ hypergeometric functions) extracted from
the Fourier-Legendre series may be summed, having values that are inverse
powers of the eight primes
$1/\left(2^{i}3^{j}5^{k}7^{l}11^{m}13^{n}17^{o}19^{p}\right)$ multiplying
powers of the coefficient $k$.
