Recently Appell systems of monogenic polynomials in R 3 were constructed by several authors. Main purpose of this paper is the description of another Appell system that is complete in the space of square integrable quaternion-valued functions. A new Taylor-type series expansion based on the Appell polynomials is presented, which can be related to the corresponding Fourier series analogously as in the complex one-dimensional case. These results find applications in the description of the hypercomplex derivative, the monogenic primitive of a monogenic function and the characterization of functions from the monogenic Dirichlet space.

The main aim of this paper is to recall the notion of the Gelfand-Tsetlin bases (GT bases for short) and to use it for an explicit construction of orthogonal bases for the spaces of spherical monogenics (i.e., homogeneous solutions of the Dirac or the generalized Cauchy-Riemann equation, respectively) in dimension 3. In the paper, using the GT construction, we obtain explicit orthogonal bases for spherical monogenics in dimension 3 having the Appell property and we compare them with those constructed by the first and the second author recently (by a direct analytic approach).

International audienceOne of the most fruitful and elegant approach (known as Kolosov-Muskhelishvili formulas) for plane isotropic elastic problems is to use two complex-valued holomorphic potentials. In this paper, the algebra of real quaternions is used in order to propose in three dimensions, an extension of the classical Muskhelishvili formulas. The starting point is the classical harmonic potential representation due to Papkovich and Neuber. Alike the classical complex formulation, two monogenic functions very similar to holomorphic functions in 2D and conserving many of interesting properties, are used in this contribution. The completeness of the potential formulation is demonstrated rigorously. Moreover, body forces, residual stress and thermal strain are taken into account as a left side term. The obtained monogenic representation is compact and a straightforward calculation shows that classical complex representation for plane problems is embedded in the presented extended formulas. Finally the classical uniqueness problem of the Papkovich-Neuber solutions is overcome for polynomial solutions by fixing explicitly linear dependencies

SUMMARYThe main goal of this paper is to construct a spatial analog to the Kolosov-Muskhelishvili formulae using the framework of the hypercomplex function theory. We prove a generalization of Goursat's representation theorem for solutions of the biharmonic equation in three dimensions. On the basis of this result, we construct explicitly hypercomplex displacement and stress formulae in terms of two monogenic functions.

The main objective of this article is to consider complete orthonormal systems of monogenic polynomials together with their hypercomplex derivatives. Desired is that the derivatives of the basis polynomials are again basis functions from the original system. Based on this result, we prove an orthogonal decomposition of the space of square integrable monogenic functions with respect to the derivatives of arbitrary order.