Abstract. In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions in a generalized quaternion algebra.

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In the paper [1] considered a new class of quaternionic mappings, socalled G-monogenic mappings. In this paper we prove analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, and the Cauchy integral formula for G-monogenic mappings.

Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the generalized quaternion algebras to a basis in the division quaternions algebra or to a basis in the coquaternions algebra and vice versa. The same result was obtained for the generalized octonion algebra. Moreover, we emphasize the applications of these results to the algebraic equations and De Moivre's formula in generalized quaternion algebras and in generalized octonion division algebras.

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The general linear quaternionic equation with one unknown and systems of linear quaternionic equations with two unknown are solved. Examples of equations and their systems are considered. Mathematics Subject Classification (2010). 15A06, 11R52.

Abstract. In this paper we investigated some properties of holomorphic functions (belonging to the kernel of the Dirac operator) defined on domains of the real Cayley-Dickson algebras. For this purpose, we study first some properties of these algebras, especially multiplication tables for certain elements of the basis. Using these properties, we provided an algorithm for constructing examples of the class of functions under consideration.