Decoupling of a system of partial difference equations with constant coefficients and application

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Abstract
Consider D multi‐variable functions, Aj(n), j=1 to D, where n stands for the evaluation point in the associated multi‐dimensional space of coordinates (n1,n2,...). Let us assume that the Aj’s satisfy a system of D linearly coupled finite difference equations: the value of each function Ai at the evaluation point n is given as a linear combination of the values of this function and others at shifted evaluation points. By introducing D suitable generating functions, Gj, j=1 to D, one is able to replace the D coupled difference equations by a system of D linear equations where the Gj’s play the role of the D unknowns. After solving this new system of equations, it is then possible to construct a difference equation for each of the Aj’s relating the value of Ai at the evaluation point n to the values of Ai itself at shifted arguments. The solution of such a decoupled equation can then be handled using the multi‐dimensional combinatorics function technique.