A generalization of Poincare's theorem for recurrence equations
Journal of Approximation Theory
An introduction to difference equations
An introduction to difference equations
Asymptotic behaviour of solutions of linear recurrences and sequences of Mo¨bius-transformations
Journal of Approximation Theory
Spectral properties of Jacobi matrices by asymptotic analysis
Journal of Approximation Theory
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In recent years many results have been obtained on the asymptotic behavior of solutions of the matrix difference equation M"nx"n=x"n"+"1 where {M"n}"n"="0^~ is a sequence of kxk-matrices with real or complex entries that are close to diagonal matrices. In this paper we study the question of how to transform a matrix sequence {M"n}"n"="0^~ where the entries behave sufficiently regularly, into a sequence of almost-diagonal matrices, so that the results for almost-diagonal matrices can be applied to the difference equation with the transformed sequence. In particular, we will try to find explicit matrices B"n such that the matrices M"n^'=B"n"+"1^-^1M"nB"n are close to diagonal matrices and a Levinson-type theorem can be applied to transform the sequence {M"n^'}"n"="0^~ into a sequence of diagonal matrices. In the case that the M"n are real 2x2-matrices, a fairly general answer is obtained and it is shown how to proceed for a given sequence {M"n}"n"="0^~. Furthermore, we prove a couple of results that are useful for the case of general order k.