The Solution of Row-Finite Linear Systems with the Infinite Gauss-Jordan   Elimination The Case of Linear Difference Equations with Variable   Coefficients
  The construction of the general solution sequence of row-finite linear systems is accomplished by implementing -ad infinitum- the Gauss-Jordan algorithm under a rightmost pivot elimination strategy. The algorithm generates a basis (finite or Schauder) of the homogeneous solution space for row-finite systems. The infinite Gaussian elimination part of the algorithm solves linear difference equations with variable coefficients of regular order, including equations of constant order and of ascending order. The general solution thus obtained can be expressed as a single Hessenbergian. 
Alexandros G. Paraskevopoulos
math.FA