Trigonometric sums and Appell sequences - a journey from Complex to Hypercomplex Function Theory

It is well known that trigonometric sums play an important role in Fourier analysis in general, but also in number theory (with relations to Goldbach’s and Waring's problems, as well as properties of Riemann’s zeta-function) or the theory of univalent functions, among other fields. Recently, Ruscheweyh and Salinas showed the relationship of a celebrated theorem of L. Vietoris (1958) about the positivity of certain sine and cosine sums with the function theoretic concept of stable holomorphic functions in the unit disc. It will be shown that the coefficient sequence which plays a crucial role in Vietoris' theorem is identically with the number sequence that characterizes generalized Appell sequences of homogeneous Clifford holomorphic polynomials in 3D.  The talk shows that the framework of Hypercomplex Function Theory  led  directly to generalizations of Vietoris' number sequence for nD. Moreover, the approach implied their representation exclusively by non-commutative generators of Clifford algebras and hypergeometric function methods allowed to establish their generating functions.