Let \(H\) denote the skew field of quaternions endowed with the usual multiplication on the standard basis \(1\), \(i\), \(j\), \(k\) extended by distributivity. G. Gentili and D. C. Struppa [Adv. Math. 216, No. 1, 279–301 (2007; Zbl 1124.30015)] showed that for a power series \(\sum_{n\in\mathbb{N}} q^n a_n\), where \(\{a_n\}_{n\in\mathbb{N}}\) and \(q\) are in \(H\), there is an \(R\) in \([0,+\infty]\) such that the series converges absolutely and uniformly on compact sets in \(B(0,R)= \{q\in H : |q|< R\}\) and \(\sigma\) diverges in \(\{q\in H : |q|> R\}\). The authors explain the difficulties in trying to define analyticity through power series of the form \(\sum(q- p)^na_n\) for \(p\in H\) and \(p\neq 0\). They introduce a distance \(\sigma\) which is not topologically equivalent to the Euclidean distance. With the usual addition operation and the multiplication operation \(*\) defined by \[ \Biggl(\sum_{n\in\mathbb{N}} q^na_n\Biggr)* \Biggl(\sum_{n\in\mathbb{N}} q^nb_n\Biggr)= \sum_{n\in\mathbb{N}} q^n \sum^n_{k=0} a_k b_{n-k}, \] the authors are able to prove that for a sequence \(\{a_n\}_{n\in\mathbb{N}}\) in \(H\) and \(R\) in \([0,+\infty]\) such that \(1/R= \limsup_{n\to\infty} |a_n|^{1/n}\), for all \(p\in H\), the series \(f(q)= \sum_{n\in\mathbb{N}} (q- p)^{* n}a_n\) converges absolutely and uniformly on compact subsets of \(\sum(p,R)\) and it does not converge at any point of \(H\setminus\overline{\sum(p, R)}\). \(R\) is then the \(\sigma\)-radius of convergence of \(f(q)\).
Two types of analyticity are defined and related. If \(\Omega\) is an open subset of \(H\), a function \(f: \Omega\to H\) is strongly analytic at \(p\) in \(\Omega\) if there is a power series \(\sum(q-p)^{*n}a_n\) converging in a neighborhood \(U\) of \(p\) in \(\Omega\) such that \(f(q)= \sum_{n\in\mathbb{N}}(q- p)^{*n}a_n\) for all \(q\) in \(U\). \(f\) is termed strongly analytic if it is strongly analytic for all \(p\) in \(\Omega\). The main sharp result in this connection is that any function defined by a power series \(\sum_{n\in\mathbb{N}} q^na_n\) which converges in \(B(0,R)\) is strongly analytic in the open set \(JA(B)= \big\{p\in H : 2|\text{Im}(p)|< R-|p|\big\}\). Since \(A(H)= H\), all quaternionic entire functions are strongly analytic in \(H\). If \(B\neq H\), \(A(B)\) is strictly contained in \(B\). The weaker notion of analyticity defined on a \(\sigma\)-open subset of \(H\) and termed \(\sigma\)-analytic produces the theorem that a power series \(\sum q^na_n\) having radius of convergence \(R\) defines a \(\sigma\)-analytic function on \(JB(0,R)\).
Analyticity of slice regular functions, recently introduced by Gentili and Struppa in [loc. cit.] and by F. Colombo et al. in [Adv. Math. 222, No. 5, 1793–1808 (2009; Zbl 1179.30052)], is studied in the last secton of the paper. Strikingly, the authors prove a quaternionic function is slice regular in a domain if and only if it is \(\sigma\)-analytic on the same domain. The paper is well-written and provides valuable insights on analyticity of quaternionic functions.