Clifford Algebra And Continued Fractions

A continued fraction may be regarded as a sequence of Mobius transformations. In the 20th century, the analytic theory of continued fractions has been developed increasingly by the work of Jones, Thron and Lisa etc. It has been applied in transcendental functions, control theory, asymptotic series and so on. In recent years, Beardon considered continued fractions by using Clifford matrices and higher dimensional Mobius transformations. Many new results have been obtained.The main aim of this dissertation is to disscuss some properties of continued fractions and Clifford algebra from algebraic point of view.Starting from the expression of an element in Clifford algebra, we establish the complex matrix representation of 4-dimensional Clifford numbers and several properties are obtained. We propose the concept of Moore-Penrose inverse of Clifford numbers and get a necessary and sufficient condition for an element in 4-dimensional Clifford algebra to be invertible. As an application, the explicit expression of the general solution of the equation axb = c in terms of a, b and c in 4-dimensional Clifford algebra C₄ is obtained.In 1965 Hillam and Thron proved a general convergence criterion for continued fractions K(a_n/b_n). That was the famous Hillam-Thron theorem. In this dissertation, by using Clifford matrix, we establish the corresponding results in Clifford continued fractions. An application is given.