Extension of the method of continuants for n-order linear difference equations with variable coefficients.

*(English)*Zbl 0596.39002*T. Cholewicki* [Some new results in the analysis of a cascade of active and various two ports. I. Application of immitance matrices. ibid. 33, 165-174 (1985)] has shows that the second-order inhomogeneous difference equation with variable coefficients ${y}_{i+2}={p}_{i+1}{y}_{i+1}+{q}_{i}{y}_{i}+{f}_{i},$ $(i=1,2,\xb7\xb7\xb7)$ is solvable in a ”closed form” by the use of continuants. The present work shows explicitly how this can be extended to analogous third and fourth order equations and indicates that the method extends similarly to the $n$-th order case

$${y}_{i+n}={a}_{i+n-1}^{\text{'}}{y}_{i+n-1}+{a}_{i+n-2}^{2}{y}_{i+n-2}+\xb7\xb7\xb7+{a}_{i}^{n}{y}_{i}+{f}_{i},$$

and to the case of vector-matrix systems. The ”closed form” is a recursive definition and involves computing large determinants.

Reviewer: L.J.Grimm

##### MSC:

39A10 | Additive difference equations |