Formal theory of irregular linear difference equations
George David Birkhoff · Acta Mathematica · 1930
this determinant.If we replace x by x--I in this relation, we obtain a like relation between yl(x), ... y,,(x).On solving for one of the dependent variables, say y,,(x), in-this relation, and elimination Of y,~(x)in the first n--I equations (I), we obtain a like equivalent system of order n--I.Proceeding successively in this way we arrive finally at an equivalent linear system of order ~ Z~ a,,(x)y~(x).