  This paper presents recent results obtained by the authors (partly in
collaboration with A. Dabrowska) concerning expansions of zonal functions on
Euclidean spheres into spherical harmonics and some applications of such
expansions for problems involving Fourier transforms of functions with
rotational symmetry. The method used to derive the expansion formula is based
entirely on differential methods and completely avoids the use of various
integral identities commonly used in this context. Some new identities for the
Fourier transform are derived and as a byproduct seemingly new recurrence
relations for the classical Bessel functions are obtained.
