Combinatorial identities in the context of hypercomplex function theory

Abstract

Recently, the authors have shown that a certain combinatorial identity in terms of generators of quaternions is related to a particular sequence of rational numbers (Vietoris' number sequence). This sequence appeared for the first time in a theorem by Vietoris (1958) and plays an important role in harmonic analysis and in the theory of stable holomorphic functions in the unit disc. We present a generalization of that combinatorial identity involving an arbitrary number of generators of a Clifford algebra. The result reveals new insights in combinatorial phenomena in the context of hypercomplex function theory.

Non-symmetric Number Triangles Arising from Hypercomplex Function Theory in $$\mathbb {R}^{n+1}$$

“…The next result about the evaluation of the alternating series (16) for n = 2, uses the relation of T k (2) to the celebrated Catalan numbers C k = 1 k+1 2k k , which already was mentioned in [5].…”

“…Formula (5) shows that generalized Vietoris numbers (3) appear also in coefficients of a special power series multiplied by a term that characterizes the space of homogeneous polynomials in higher dimension. Indeed,…”

“…and for n = 1 formula (5) is the sum of the ordinary geometric series due to the special values of γ and δ. Since the geometric series in one real respectively one complex variable coincide (in the latter case (5) has to be considered as a Taylor series expansion in the unit complex disc) our hypercomplex approach contains the complex case as particular case R 1+1 ∼ = C. The important role of ordinary Vietoris numbers, corresponding to n = 2 in (3), in the theory of orthogonal polynomials is discussed in [1]. The paper of Ruschewey and Salinas [12] shows the relevance of the celebrated result of Vietoris for a complex function theoretic result in the context of subordination of analytic functions.…”

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

“…(51) ( 2 )2 and by the corresponding series, the left hand side and the right hand side can be written, . The comparison of the two sides leads to 0…”

“…Cação et al51 study examples of combinatorial identities in terms of generators of  0, and symmetric products of them inspired by expressions like in(28).…”