A new series expansion for slice regular functions

Abstract

A promising theory of quaternion-valued functions of one quaternionic variable, now called slice regular functions, has been introduced by Gentili and Struppa in 2006. The basic examples of slice regular functions are the power series of type ∑n∈Nqnan on their balls of convergence B(0,R)={q∈H:|q|<R}. Conversely, if f is a slice regular function on a domain Ω⊆H then it admits at each point q0∈Ω an expansion of type f(q)=∑n∈N(q−q0)∗nan where (q−q0)∗n denotes the nth power of q−q0 with respect to an appropriately defined multiplication ∗. However, the information provided by such an expansion is somewhat limited by a fact: if q0 does not lie on the real axis then the set of convergence of the series in the previous equation needs not be a Euclidean neighborhood of q0. 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.