A Pincherle-Type Convergence Theorem for Generalized Continued Fractions in Banach Algebras

This contribution is dedicated to the interdependence of higher order linear difference equations and generalized continued fractions in Banach algebras. It turns out that the computation of certain subdominant solutions of a higher order linear difference equation can be done more efficiently by considering its adjoint equation. The final result is a Pincherle-type convergence criterion for generalized continued fractions in terms of the adjoint equation. To demonstrate the applicability of the method, it is applied to the Poincaré–Perron difference equation and selected problems from pure mathematics and applied stochastic processes.