On a Solvable Symmetric and a Cyclic System of Partial Difference Equations

Stevo Stević, Bratislav Iričanin, Zdenĕk Šmarda

It is shown that the following symmetric system of partial difference equations c m,n = d m−1,n + c m−1,n−1, d m,n = c m−1,n + d m−1,n−1 , is solvable on the combinatorial domain C = (m, n) ∈ N 2 0 : 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 c j,j, c j,0, d j,j, d j,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.