On polynomial series expansions of Cliffordian functions

Abstract

Communicated by W. SprößigClassical results on the expansion of complex functions in a series of special polynomials (namely inverse similar sets of polynomials) are extended to the Clifford setting. This expansion is shown to be valid in closed balls.

Representation in Fréchet Spaces of Hyperbolic Theta and Integral Operator Bases for Polynomials

“…1. Previous studies [8,21,32,37,38,42,43] Bessel, Hermite, and Gontcharoff polynomials to hypercomplex analysis is a potential avenue. Moreover, the investigation into the effectiveness, growth type, and order of the above sets could be extended to several complex variables, covering regions such as polycylindrical, hyperelliptical, spherical, and Faber regions, for all entire functions and at the origin.…”

On Hadamard's three‐hyperballs theorem and its applications to Whittaker‐Cannon hypercomplex theory

“…Then, △ k converges absolutely for all k. Moreover, given 𝜖 > 0, we choose N such that ∑ ∞ n=N |a n |F n (R) < 𝜖; hence, by (11), we have ∑ ∞ n=N |a n |R n < 𝜖. Draw a row line through 𝜋 N,0 in the matrix Π, and a column line so that the elements in the top right-hand corner are zero, say through…”

“…Most relevant to our study are the connections between the special monogenic polynomials and power series expansions and the flexibility afforded by those approaches. [9][10][11] Much of the older theory of special monogenic polynomials has been given a different interpretation. A new light has been shed upon the study of elementary functions, [12][13][14][15][16][17][18] the computation of combinatorial identities, [19][20][21] and the study of a generalized Joukowski transformation in Euclidean space of arbitrary higher dimension.…”

“…Abul‐Ez et al 9 proved a counterpart of Hadamard's three‐hyperballs theorem within the context of hypercomplex analysis and studied the overconvergence properties of a monogenic basic series associated with the prescribed special monogenic polynomials. Most relevant to our study are the connections between the special monogenic polynomials and power series expansions and the flexibility afforded by those approaches 9–11 . Much of the older theory of special monogenic polynomials has been given a different interpretation.…”

On the representation of monogenic functions by the product bases of polynomials

“…From this starting point, many results on the polynomial bases in the complex case of one complex variable were refined and generalized to the Clifford setting (see [1][2][3][4][5][6][7][8][9]27]). In this line of research in Clifford setting, one of the interesting problem has been investigated by Zayed et al [32] where the authors explored the effectiveness of the hypercomplex derivative and primitive basic sets associated with the previously mentioned polynomials.…”