In [Gentili, G. and Stoppato, C., 2007, Zeros of regular functions and polynomials of a quaternionic variable, n. 1 (Dipartimento di Matematica “U. Dini”, Universita di Firenze), Preprint; Gentili, G. and Struppa, D.C., 2006, A new approach to Cullen-regular functions of a quaternionic variable. C. R. Acad. Sci. Paris, Ser. I, 342, 741–744; Gentili, G. and Struppa, D.C., 2007, A new theory of regular functions of a quaternionic variable. Adv. Math., 216, 279–301.] the authors offered an alternative definition and theory of regularity for functions of a quaternionic or octonionic variable, inspired by an idea of Cullen [Cullen, C.G., 1965, An integral theorem for analytic intrinsic functions on quaternions. Duke Mathematics Journal, 32, 139–148]. This alternative theory is intriguing because it allows the study of natural power series (and polynomials) with quaternionic or octonionic coefficients, which is excluded when the Fueter approach is followed. In this article the authors follow the same ideas to o...