Hypercomplex function theory generalizes the theory of holomorphic functions of one complex variable by using Clifford Algebras and provides the fundamentals of Clifford Analysis as a refinement of Harmonic Analysis in higher dimensions. We define the Laguerre derivative operator in hypercomplex context and by using operational techniques we construct generalized hypercomplex monogenic Laguerre polynomials. Moreover, Laguerre-type exponentials of order m are defined.

We construct a family of hypercomplex Appell polynomials and it corresponding generating function of exponential type. An extension of this approach leads to the construction of a complete set of monogenic Appell polynomials. Moreover, it is shown how the obtained Appell family is connected with complete sets of solutions of the Riesz system.

This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence S used in a celebrated theorem on positive trigonometric sums by L. Vietoris in 1958. For S a generating function is obtained which allows to derive an interesting relation to a result deduced in 1974 by Askey and Steinig about some trigonometric series. The further study of S is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.Mathematics Subject Classification (2010). 30G35;11B83;05A10.

This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows to prove their recursive construction in analogy to the complex power functions. This property can somehow be considered as a compensation for the loss of multiplicativity caused by the non-commutativity of the underlying algebra.

As it is well-known, the generalization of the classical CauchyRiemann system to higher dimensions leads to the so-called Riesz and MoisilTeodorescu systems. Rewriting these systems in quaternionic language and taking advantage of the underlying algebra, we construct complete sets of polynomials solutions of both systems that are orthonormal with respect to a certain inner product. The restrictions of those polynomials to the unit sphere can be viewed as analogues to the complex case of the Fourier exponential functions {e inθ } n≥0 on the unit circle and constitute a refinement of the wellknown spherical harmonics.