On linear differential equations with variable coefficients involving a para-Grassmann variable

Abstract

Linear differential equations with constant coefficients involving a para-Grassmann variable have been considered recently in the work of Mansour and Schork [Symmetry, Integr. Geom.: Methods Appl. 5, 73 (2009)]. In the present paper, this treatment is extended to linear differential equations with variable coefficients. For the equation of first order, an explicit formula for the solution is given. For the equations of higher order, it is shown how the solutions may be determined in terms of the solutions of “ordinary” differential equations (i.e., involving only “bosonic” variables). Some examples of these differential equations are discussed and analogs for the trigonometric functions are introduced.