Solution formulas for linear difference equations with applications to continued fractions (Q1330857)

In the space of \(q\)-dimensional complex matrices a linear difference equation (1) \(X_ n = A_ n X_{n - 1} + U_ n\), \(n \in N\), is considered. Formulas connecting solutions of equation (1) with solutions of some homogeneous equation from the neighborhood of equation (1) are constructed and asymptotic estimates of solutions of (1) are obtained. In the second part of the paper the author investigates the second-order linear difference equation \(x_ n = a_ 1 (n)x_{n - 1} + a_ 2(n) x_{n - 2}\), \(x_ n \in C\), \(a_ 2(n) \neq 0\), \(n \in N\), and the corresponding continued fraction \(K^ \infty_{n = 1} (a_ 2(n)/a_ 1(n))\). Using results of the first part conditions of convergence of the continued fraction \(K^ \infty_{n = 1} (a_ 2(n)/a_ 1(n))\) in terms of convergence of a continued fraction from its neighborhood are established and approximation formulas are given. Analogous theorems for continued fractions analytically depending on a parameter from some Riemann surface are established.