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Leibniz' combinatorial formula for determinants is modified to establish acondensed and easily handled compact representation for Hessenbergians,referred to here as Leibnizian representation. Alongside, the elements of afundamental solution set associated with linear difference equations withvariable coefficients of order $p$ are explicitly represented by $p$ bandedHessenbergian solutions, built up solely of the variable coefficients. Thisyields banded Hessenbergian representations for the elements both of theproduct of companion matrices and of the determinant ratio formula of theone-sided Green's function (Green's function for short). Combining the aboveresults, the elements of the foregoing notions are endowed with compactrepresentations formulated here by Leibnizian and nested sum representations.We show that the elements of the fundamental solution set can be expressed interms of the first banded Hessenbergian fundamental solution, called principaldeterminant function. We also show that the Green's function coincides with theprincipal determinant function, when both functions are restricted to a fairlylarge domain. These results yield, an explicit and compact representation ofthe Green's function restriction along with an explicit and compact solutionrepresentation of the previously stated type of difference equations in termsof the variable coefficients, the initial conditions and the forcing term. Theequivalence of the Green's function solution representation and the well knownsingle determinant solution representation is derived from first principles.Algorithms and automated software are employed to illustrate the main resultsof this paper.

Explicit and compact representations for the Green's function and the Solution of Linear Difference Equations with variable coefficients