  A promising theory of quaternion-valued functions of one quaternionic
variable, now called slice regular functions, has been introduced in 2006. The
basic examples of slice regular functions are power series centered at 0 on
their balls of convergence. Conversely, if f is a slice regular function then
it admits at each point of its domain an expansion into power series, where the
powers are taken with respect to an appropriately defined multiplication *.
However, the information provided by such an expansion is somewhat limited by a
fact: if the center p of the series does not lie on the real axis then the set
of convergence needs not be a Euclidean neighborhood of p. We are now able to
construct a new type of expansion that is not affected by this phenomenon: an
expansion into series of polynomials valid in open subsets of the domain. Along
with this construction, we present applications to the computation of the
multiplicities of zeros and of partial derivatives.
