A closed form to the general solution of linear difference equations   with variable coefficients
  The determinant of a lower Hessenberg matrix (Hessenbergian) is expressed as a sum of signed elementary products indexed by initial segments of nonnegative integers. A closed form alternative to the recurrence expression of Hessenbergians is thus obtained. This result further leads to a closed form of the general solution for regular order linear difference equations with variable coefficients, including equations of N-order and equations of ascending order. 
A. G. Paraskevopoulos and M. Karanasos
math.FA