  A novel method, connecting the space of solutions of a linear differential
equation, of arbitrary order, to the space of monomials, is used for exploring
the algebraic structure of the solution space. Apart from yielding new
expressions for the solutions of the known differential equations, the
procedure enables one to derive various properties of the orthogonal
polynomials and functions, in a unified manner. The method of generalization of
the present approach to the multi-variate case is pointed out and also its
connection with the well-known factorization technique. It is shown that, the
generating functions and Rodriguez formulae emerge naturally in this method.
