Linking Clifford analysis and combinatorics through bijective methods

Abstract

The application of Clifford Analysis methods in Combinatorics has some peculiarities due to the use of noncommutative algebras. But it seems natural to expect from here some new results different from those obtained by using approaches based on several complex variable. For instances, the fact that in Clifford Analysis the point-wise multiplication of monogenic functions as well as their composition are not algebraically closed in this class of generalized holomorphic functions causes serious problems. Indeed, this is one of the reasons why in polynomial approximation almost every problem needs the construction of specially adapted polynomial bases. Our aim is to show that the analysis and comparison of different representations of the same polynomial or entire function allow to link Clifford Analysis and Combinatorics by means of bijective methods. In this context we also stress the central role of the hypercomplex derivative for power series representations in connection with the concept of Appell sequences as analytic tools for establishing this link.