Linear differential and difference systems: EG δ - and EG σ -eliminations. Systems of linear ordinary differential and difference equations of the form A r (x)ξ r y(x)+⋯+A 1 (x)ξy(x)+A 0 (x)y(x)=0, ξ∈d dx , E, where E is the shift operator, Ey(x)=y(x+1), are considered. The coefficients A i (x), i=0,⋯,r, are square matrices of order m, and their entries are polynomials in x over a number field K, with A r (x) and A 0 (x) being nonzero matrices. The equations are assumed to be independent over K[x,ξ]. For any system S of this form, algorithms EG δ (in the differential case) and EG σ (in the difference case) construct, in particular, the l-embracing system S ¯ of the same form. The determinant of the leading matrix A ¯ r (x) of this system is a nonzero polynomial, and the set of solutions of system S ¯ contains all solutions of system S. (Algorithm EG δ provides also a number of additional possibilities). Examples of problems that can be solved with the help of EG δ and EG σ are given. The package EG implementing the proposed algorithms in Maple is described.

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