86 documents found, page 1 of 9
On the Pascal simplex with hypercomplex entries
Cruz, Carla; Falcão, M. I.; Malonek, H.; Tomaz, G.We propose a generalization of the Pascal $n$-simplex by considering the generators $e_1,e_2,\dots,e_n$ of the $2^n-$dimensional Clifford algebra ${\mathcal{C} \ell}_{0,n}$ over $\mathbb{R}$in the multinomial expansion of powers of their sum. We investigate various patterns within this structure and examine several of its properties and associated combinatorial identities. Our results establish a direct connect...
Relating certain coefficients of special hyperholomorphic polynomials with a ho...
Cação, I.; Malonek, H. R.; Falcão, M. I.; Tomaz, G.Hyperholomorphic polynomials are generalizations of holomorphic polynomials to higher dimensions in the framework of hypercomplex function theory. We consider a special sequence of hyperholomorphic polynomials whose coefficients can be arranged as triangular arrays of the Pascal type. Our main focus is a one-parameter sequence formed by partial alternating sums of their main diagonal elements that in the limit ...
Generalized Vietoris’ number sequences from real and hypercomplex points of view
Cação, I.; Malonek, H. R.; Falcão, M. I.; Tomaz, G.We revisit a special rational number sequence, introduced by L. Vietoris in 1958 in the study of the positivity of some trigonometric sums and used in other contexts by several authors. The aim of the present paper is to embrace and explore real and hypercomplex analytical methods to obtain generalizations of that rational number sequence, where Jacobi polynomials and generalized Appell polynomials are involved.
Intrinsic properties of a non-symmetric number triangle
Cação, Isabel; Malonek, Helmuth R.; Falcão, M. I.; Tomaz, GraçaSeveral authors are currently working on generalized Appell polynomials and their applications in the framework of hypercomplex function theory in Rn+1. A few years ago, two of the authors of this paper introduced a prototype of these generalized Appell polynomials, which heavily draws on a one-parameter family of non-symmetric number triangles T (n), n ≥ 2. In this paper, we prove several new and interesting p...
The stability of complex dynamics for two families of coquaternionic quadratic ...
Falcão, M. I.; Miranda, Fernando; Severino, Ricardo; Soares, M. J.In this work, we begin by demonstrating that attractors, both periodic and aperiodic, of the one-parameter family of complex quadratic maps x2+ c, where c is a complex number, maintain their stability when we transition from the complex plane C to the coquaternions Hcoq as the map’s phase space. Next, we investigate the same question for a different family of quadratic maps, x2+ bx, and find that this is not th...
Remarks on the Vietoris sequence and corresponding convolution formulas
Cação, Isabel; Falcão, M. I.; Malonek, Helmuth R.; Miranda, Fernando; Tomaz, GraçaIn this paper we consider the so-called Vietoris sequence, a sequence of rational numbers of the form ck=12k(k⌊k2⌋), k= 0, 1, ⋯. This sequence plays an important role in many applications and has received a lot of attention over the years. In this work we present the main properties of the Vietoris sequence, having in mind its role in the context of hypercomplex function theory. Properties and patterns of the c...
Dynamics of the coquaternionic maps x^2 + bx
Falcão, M. I.; Miranda, Fernando; Severino, Ricardo; Soares, M. J.This paper deals with the dynamics of the one-parameter family of coquaternionic quadratic maps x2+ bx. By making use of recent results for the zeros of one-sided coquaternionic polynomials, the fixed points are analytically determined. The stability of these fixed points is also addressed, where, in some cases, due to the appearance of sets of non-isolated points, a suitably adapted notion of stability is used...
Starting with the differential: representation of monogenic functions by polyno...
Malonek, H. R.; Cação, I.; Falcão, M. I.; Tomaz, G.This paper deals with different power series expansions of generalized holomorphic (monogenic) functions in the setting of Clifford Analysis. Our main concern are generalized Appell polynomials as a special class of monogenic polynomials which have been introduced in 2006 by two of the authors using several monogenic hypercomplex variables. We clarify the reasons why a particular pair of non-monogenic variables...
A Two-step Quaternionic Root-finding Method
Falcão, M. I.; Miranda, Fernando; Severino, Ricardo; Soares, M. J.In this paper we present a new method for determining simultaneously all the simple roots of a quaternionic polynomial. The proposed algorithm is a two-step iterative Weierstrass-like method and has cubic order of convergence. We also illustrate a variation of the method which combines the new scheme with a recently proposed deflation procedure for the case of polynomials with spherical roots.
Generalized Vietoris’ number sequences from real and hypercomplex points of view
Cação, I.; Malonek, H. R.; Falcão, M. I.; Tomaz, G.We revisit a special rational number sequence, introduced by L. Vietoris in 1958 in the study of the positivity of some trigonometric sums and used in other contexts by several authors. The aim of the present paper is to embrace and explore real and hypercomplex analytical methods to obtain generalizations of that rational number sequence, where Jacobi polynomials and generalized Appell polynomials are involved.