On the structure of generalized Appell sequences of paravector valued homogeneous monogenic polynomials

Abstract

The fact that generalized Appell sequences of monogenic polynomials in the setting of hypercomplex function theory also satisfy a corresponding binomial type theorem allows to obtain their explicit structure. Recently it has been obtained a complete characterization in the case of paravector valued homogeneous polynomials of three real variables. The aim of this contribution is the study of paravector valued homogeneous polynomials of four real variables, where new types of generalized Appell sequences could be detected.

Monogenic Polynomials of Four Variables with Binomial Expansion

“…For the sake of better understanding, we reproduce here the main result of [10]. From now on we use briefly…”

“…Later on, we called attention in several papers ( [10,11,12,32]) to the relationship of Appell polynomials with an old concept of totally regular variables, first mentioned by R. Delanghe in [14]. Moreover, we discussed also the possibility of constructing pseudo-complex bases of pseudo-complex powers (see Sect.…”

“…In [10] the authors started to extend this result by considering the corresponding n = 3 form of the polynomials (10), i. e.,…”

“…In this work we are particularly concerned with the proof of a conjecture in [10] about the coefficients of the third type of Appell polynomials in four variables. Indeed, it might be that an increasing number of variables implies an increasing number of new types of generalized Appell polynomials unknown so far.…”

On Numerical Aspects of Pseudo-Complex Powers in ℝ3

“…The authors initiated recently ( [7,8,9,21]) the use of PCP as building blocks for Clifford-valued homogeneous polynomials in the Clifford Algebra C 0,n . The main differences to other approaches (cf.…”