On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space - Delanghe,Richard (Universidad de La Frontera. Departamento de Matemática y Estadística., 2010)
Let for s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 be the space of s-vectors in the Clifford algebra IR0,m+1 constructed over the quadratic vector space IR0,m+1 and let r, p, q, ∈ IN be such that 0 ≤ r ≤ m + 1, p < q and r + 2q ≤ m + 1. The associated linear system of first order partial differential equations derived from the equation ∂xW = 0 where W is IR(r,p,q)0,m+1 = ∑q j=p ⊕IR(r+2j)0,m+1 -valued and ∂x is the Dirac operator in IRm+1, is called a generalized Moisil-Théodoresco system of type (r, p, q) in IRm+1. For k ∈ N, k ≥ 1,MT+(m+ 1; k; IR(r,p,q)0,m+1), denotes the space of IR(r,p,q)0,m+1-valued homogeneous polynomials Wc of degree k in IRm+1 satisfying ∂xWx = 0. A characterization of Wk∈ MT+(m + 1; k;IR(r,p,q)0,m+1) is given in terms of a harmonic potential Hk+1 belonging to a subclass of IR(r,p,q)0,m -valued solid harmonics of degree (k + 1) in IRm+1. Furthermore, it is proved that each Wk∈ MT+(m+ 1; k; IR(r,p,q)0,m+1) admits a primitive Wk+1 ∈ MT+(m+ 1; k + 1; IR(r,p,q)0,m+1). Special attention is paid to the lower dimensional cases IR³ and IR4. In particular, a method is developed for constructing bases for the spaces MT+(4; k; IR(r,p,q)0,4), r being even.