Vietoris' number sequence and its generalizations through hypercomplex function theory

Abstract

The so-called Vietoris' number sequence is a sequence of rational numbers that appeared for the rst time in a celebrated theorem by Vietoris (1958) about the positivity of certain trigonometric sums with important applications in harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic functions (Ruscheweyh/ Salinas, 2004). In the context of hypercomplex function theory those numbers appear as coe cients of special homogeneous polynomials in R3 whose generalization to an arbitrary dimension n lead to a n-parameter generalized Vietoris' number sequence that characterizes hypercomplex Appell polynomials in Rn.