## Helmuth R. Malonek

## On generalized Vietoris' number sequences – from positivity of trigonometric sums to Appell sequences in 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 to properties of Riemann’s zeta-function) and in 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. Here, it will be shown that the coefficient sequence which plays a crucial role in Vietoris' theorem is identically to 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 leads directly to generalizations of Vietoris' number sequence to the nD-case. Moreover, the approach implies the representation of these generalized sequences exclusively by non-commutative generators of Clifford algebras and hypergeometric function methods allow to establish their generating functions.

Joint work with I. Cação (University of Aveiro) and M. I. Falcão (University of Minho).

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. Here, it will be shown that the coefficient sequence which plays a crucial role in Vietoris' theorem is identically to 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 leads directly to generalizations of Vietoris' number sequence to the nD-case. Moreover, the approach implies the representation of these generalized sequences exclusively by non-commutative generators of Clifford algebras and hypergeometric function methods allow to establish their generating functions.

Joint work with I. Cação (University of Aveiro) and M. I. Falcão (University of Minho).

Milton Ferreira

**Fundamental Solutions for some Multidimensional Fractional ****differential operators**

Recently there is a surge in interest in PDEs involving fractional derivatives in different fields of

engineering, such as mechanics, watermarking, etc.. In this talk we will present, by application of

transform methods, fundamental solutions for some multidimensional fractional differential opera-

tors: the fractional Laplace and Dirac operators defined by fractional Riemann-Liouville derivatives

or Caputo derivatives and the time-fractional heat and parabolic Dirac operators.

Joint work with Nelson Vieira (University of Aveiro).

engineering, such as mechanics, watermarking, etc.. In this talk we will present, by application of

transform methods, fundamental solutions for some multidimensional fractional differential opera-

tors: the fractional Laplace and Dirac operators defined by fractional Riemann-Liouville derivatives

or Caputo derivatives and the time-fractional heat and parabolic Dirac operators.

Joint work with Nelson Vieira (University of Aveiro).