Monogenic pseudo-complex power functions and their applications

Abstract

The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D-elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan-Sofo type binomial identity is also proved. Information

Monogenic Polynomials of Four Variables with Binomial Expansion

“…Numerical advantages of our approach to the 3D-case are discussed in our second contribution to this conference: On Numerical Aspects of PseudoComplex Powers in R 3 . Finally, the recent paper [13] shows that pseudo-complex powers are also relevant for combinatorial problems and open new fields for applications of HFT.…”

“…We can assume that a 1 a 2 a 3 = 0, since the trivial case where one of the a i 's is zero was already considered in (13). The solution of (14) is:…”

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.…”

“…Among other problems, we discussed in [9] the applications of PCP to some problems of combinatorial nature. In [21] we studied their role in 3D Appell systems and extended the obtained results to the 4D quaternionic case in [8].…”

“…From the other side, it led us to interesting studies about different parameter choices and its closed connection with the primitive roots of unity as mentioned before. Moreover, the freedom of the parameter choice implies that PCP can play an important role for the application of bijective methods from enumerative and algebraic combinatorics (see [9]). …”