On a solvable symmetric and a cyclic system of partial difference equations

Abstract

It is shown that the following symmetric system of partial difference equations cm,n = dm-1,n + cm-1, n-1, dm,n = cm-1,n + dm-1,n-1, is solvable on the combinatorial domain C = n {(m,n) ? N20 : 0 ? n ? m}\{(0,0)}, by presenting some formulas for the general solution to the system on the domain in terms of the boundary values cj,j, cj,0, dj,j, dj,0, j ? N, and the indices m and n. The corresponding result for a related three-dimensional cyclic system of partial difference equations is also proved. These results can serve as a motivation for further studies of the solvability of symmetric, close-to-symmetric, cyclic, close-to-cyclic and other related systems of partial difference equations.