Linear differential and difference systems: EGδ- and EGσ- eliminations

• Authors: S. A. Abramov;D. E. Khmelnov
• Venue: Programming and Computing Software
• Year: 2013

Systems of linear ordinary differential and difference equations of the form $$A_r (x)\xi ^r y(x) + \ldots + A_1 (x)\xi y(x) + A_0 (x)y(x) = 0,\xi \in \left\{ {\frac{d} {{dx}},E} \right\}$$, 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 EGδ (in the differential case) and EGσ (in the difference case) construct, in particular, the l-embracing system $$\bar S$$ of the same form. The determinant of the leading matrix $$\bar A_r (x)$$ of this system is a nonzero polynomial, and the set of solutions of system $$\bar S$$ contains all solutions of system S. (Algorithm EGδ provides also a number of additional possibilities.) Examples of problems that can be solved with the help of EGδ and EGσ are given. The package EG implementing the proposed algorithms in Maple is described.