Generalized Clifford Algebras and the N -th Root of Linear Differential Equations of Higher Order

It is shown how generalized Clifford algebras allows to construct the N-th root of N-order linear differential equations involving massless and massive particles. The N-th higher-order linear differential equation is equivalent, after a factorization and cyclic permutation of the factors, to N first-order differential equations. Explicit solutions are found. The study of the generalized Dirac equation and gauge theories in Generalized Clifford Spaces follow. We conclude with some remarks on pseudo-unitary algebras, modular arithmetic, modified Dirac equations, ordinary Clifford algebras, Octonions, and the Okubo algebra.