On generalized Vietoris’ number sequences

Abstract

Recently, by using methods of hypercomplex function theory, the authors have shown that a certain sequence S of rational numbers (Vietoris' sequence) combines seemingly disperse subjects in real, complex and hypercomplex analysis. This sequence appeared for the first time in a theorem by Vietoris (1958) with important applications in harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic functions (Ruscheweyh/Salinas, 2004). A non-standard application of Clifford algebra tools for defining Clifford-holomorphic sequences of Appell polynomials was the hypercomplex context in which a one-parametric generalization S(n), n ≥ 1, of S (corresponding to n = 2) surprisingly showed up. Without relying on hypercomplex methods this paper demonstrates how purely real methods also lead to S(n). For arbitrary n ≥ 1 the generating function is determined and for n = 2 a particular case of a recurrence relation similar to that known for Catalan numbers is proved.

A Pascal-Like Triangle with Quaternionic Entries

“…A similar result was proved in [4] by using the T. Abadie's formula for the derivative of a composed function. In its present form, (9) can be used to easily identify several properties of the quaternionic Pascal triangle.…”

A Sturm‐Liouville equation on the crossroads of continuous and discrete hypercomplex analysis

“…In Cação et al 34 the authors proved that Vietoris' numbers ( ) can be generated via the Gauss' hypergeometric function.…”

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

“…[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. 22 Earlier results in the theory of special polynomial bases in hypercomplex analysis and its counterpart in the function theory of several complex variables can be found in previous studies 12,[23][24][25][26][27][28][29][30][31][32][33][34] and elsewhere.…”