On generalized Vietoris' number sequences - origins, properties and applications

TL;DR

This paper explores generalized Vietoris' number sequences, revealing their origins, properties, and applications within hypercomplex function theory, connecting classical sine and cosine sum positivity with Clifford holomorphic polynomials.

Contribution

It introduces one-parameter generalizations of Vietoris' sequences and demonstrates their significance in hypercomplex function theory and Clifford analysis.

Findings

Sequences are identical with those characterizing generalized Appell sequences.

Properties of the sequences are studied in the context of hypercomplex functions.

Applications include insights into positivity of sine and cosine sums and Clifford holomorphic polynomials.

Abstract

Ruscheweyh and Salinas showed in 2004 the relationship of a celebrated theorem of Vietoris (1958) about the positivity of certain sine and cosine sums with the function theoretic concept of stable holomorphic functions in the unit disc. The present paper shows that the coefficient sequence in Vietoris' theorem is identical with the number sequence that characterizes generalized Appell sequences of homogeneous Clifford holomorphic polynomials in $\mathbb{R}^3.$ The paper studies one-parameter generalizations of Vietoris' number sequence, their properties as well as their role in the framework of Hypercomplex Function Theory.