Matrix Continued Fractions Related to First-order Linear Recurrence Systems∗

We introduce a matrix continued fraction associated with the first-order linear recurrence system Yk = θkYk−1. A Pincherle type convergence theorem is proved. We show that the n-th order linear recurrence relation and previous generalizations of ordinary continued fractions form a special case. We give an application for the numerical computation of a non-dominant solution and discuss special cases where θk is constant for all k and the limiting case where limk→+∞ θk is constant. Finally the notion of adjoint fraction is introduced which generalizes the notion of the adjoint of a recurrence relation of order n.