  Given a quaternionic slice regular function $f$, we give a direct and
effective way to compute the coefficients of its spherical expansion at any
point. Such coefficients are obtained in terms of spherical and slice
derivatives of the function itself. Afterwards, we compare the coefficients of
$f$ with those of its slice derivative $\partial_{c}f$ obtaining a countable
family of differential equations satisfied by any slice regular function. The
results are proved in all details and are accompanied to several examples.
  For some of the results, we also give alternative proofs.
