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
$$c_{m,n}=d_{m-1,n}+c_{m-1,n-1},$$
$$d_{m,n}=c_{m-1,n}+d_{m-1,n-1},$$
is solvable on the combinatorial domain ${\mathcal C}=\big\{(m,n)\in N^2_0 : 0\le n\le m\big\}\setminus\{(0,0)\},$ by presenting some formulas for the general solution to the system on the domain in terms of the boundary values $c_{j,j},$ $c_{j,0},$ $d_{j,j},$ $d_{j,0},$ $j\in N,$ and the indices $m$ and $n$. The corresponding result for a related three-dimensional cyclic system of partial difference equations is also proved. As far as we know, these are the first results of this type in the literature and this paper initiates the study of the solvability of symmetric, close-to-symmetric, cyclic, close-to-cyclic and other related systems of partial difference equations.