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

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.