TY - CHAP A1 - Malonek, Helmuth Robert ED - Gürlebeck, Klaus ED - Könke, Carsten T1 - SPECIAL FUNCTIONS VERSUS ELEMENTARY FUNCTIONS IN HYPERCOMPLEX FUNCTION THEORY N2 - In recent years special hypercomplex Appell polynomials have been introduced by several authors and their main properties have been studied by different methods and with different objectives. Like in the classical theory of Appell polynomials, their generating function is a hypercomplex exponential function. The observation that this generalized exponential function has, for example, a close relationship with Bessel functions confirmed the practical significance of such an approach to special classes of hypercomplex differentiable functions. Its usefulness for combinatorial studies has also been investigated. Moreover, an extension of those ideas led to the construction of complete sets of hypercomplex Appell polynomial sequences. Here we show how this opens the way for a more systematic study of the relation between some classes of Special Functions and Elementary Functions in Hypercomplex Function Theory. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Architektur KW - Computerunterstütztes Verfahren KW - Computer Science Models in Engineering; Multiscale and Multiphysical Models; Scientific Computing Y1 - 2010 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170314-28702 UR - http://euklid.bauing.uni-weimar.de/ikm2009/paper.html SN - 1611-4086 ER - TY - CHAP A1 - Malonek, Helmuth Robert A1 - Falcão, M. Irene A1 - Cruz, Carla ED - Gürlebeck, Klaus ED - Lahmer, Tom ED - Werner, Frank T1 - TOTALLY REGULAR VARIABLES AND APPELL SEQUENCES IN HYPERCOMPLEX FUNCTION THEORY T2 - Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 04 - 06 2012, Bauhaus-University Weimar N2 - The aim of our contribution is to clarify the relation between totally regular variables and Appell sequences of hypercomplex holomorphic polynomials (sometimes simply called monogenic power-like functions) in Hypercomplex Function Theory. After their introduction in 2006 by two of the authors of this note on the occasion of the 17th IKM, the latter have been subject of investigations by different authors with different methods and in various contexts. The former concept, introduced by R. Delanghe in 1970 and later also studied by K. Gürlebeck in 1982 for the case of quaternions, has some obvious relationship with the latter, since it describes a set of linear hypercomplex holomorphic functions all power of which are also hypercomplex holomorphic. Due to the non-commutative nature of the underlying Clifford algebra, being totally regular variables or Appell sequences are not trivial properties as it is for the integer powers of the complex variable z=x+ iy. Simple examples show also, that not every totally regular variable and its powers form an Appell sequence and vice versa. Under some very natural normalization condition the set of all para-vector valued totally regular variables which are also Appell sequences will completely be characterized. In some sense the result can also be considered as an answer to a remark of K. Habetha in chapter 16: Function theory in algebras of the collection Complex analysis. Methods, trends, and applications, Akademie-Verlag Berlin, (Eds. E. Lanckau and W. Tutschke) 225-237 (1983) on the use of exact copies of several complex variables for the power series representation of any hypercomplex holomorphic function. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Computerunterstütztes Verfahren Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170314-27751 UR - http://euklid.bauing.uni-weimar.de/ikm2012 SN - 1611-4086 ER -