Universidade do Minho DMA Departamento de Matemática e Aplicações Universidade do Minho Campus de Gualtar 4710-057 Braga Portugal www.math.uminho.pt Universidade do Minho Escola de Ciências Departamento de Matemática e Aplicações On Generalized Hypercomplex Laguerre-type Exponentials and Applications I.Ca¸c˜ ao a M.I. Falc˜ ao b H.R. Malonek a a Departamento de Matem´ atica, Universidade de Aveiro, Portugal b Departamento de Matem´ atica e Aplica¸ c˜ oes, Universidade do Minho, Portugal Information Keywords: Hypercomplex Laguerre derivative, Appell sequences, exponential operators, functions of hypercomplex vari- ables. Original publication: Computational Science and its Applications, Lecture Notes in Computer Science, vol. 6784, pp. 271-286, 2011 DOI: 10.1007/978-3-642-21931-3 22 www.springerlink.com Abstract In hypercomplex context, we have recently con- structed Appell sequences with respect to a gen- eralized Laguerre derivative operator. This con- struction is based on the use of a basic set of monogenic polynomials which is particularly easy to handle and can play an important role in ap- plications. Here we consider Laguerre-type expo- nentials of order m and introduce Laguerre-type circular and hyperbolic functions. 1 Introduction Hypercomplex function theory, renamed Clifford Analysis ([3]) in the 1980s, when it grew into an autonomous discipline, studies functions with values in a non-commutative Clifford Algebra. It has its roots in quaternionic analysis, developed mainly in the third decade of the 19th century ([12, 13]) as another generalization of the classical theory of functions of one complex variable compared with the theory of functions of several complex variables. Curiously, but until the end of the 1990s the dominant opinion was that in Clifford Analysis only the generalization of Riemann’s approach to holomorphic functions as solutions of the Cauchy-Riemann differen- tial equations allows to define a class of generalized holomorphic functions of more than two real variables, suitable for applications in harmonic analysis, boundary value problems of partial differential equations and all the other classical fields where the theory of functions of one complex variable plays a prominent role. Logical, the methods employed during this period relied essentially on integral representations of those gener- alized holomorphic functions as consequence of the hypercomplex form of Stokes’ integral formula, including series representations obtained by the development of the hypercomplex harmonic Cauchy kernel in series of Gegenbauer polynomials, for instance ([3]).