The paper deals with the higher-order ordinary differential equations and the analogous higher-order difference equations and compares the corresponding fundamental concepts. Important dissimilarities appear for the moving frame method.
The second order linear difference equation where and , is considered as a special type of symplectic systems. The concept of the phase for symplectic systems is introduced as the discrete analogy of the Borůvka concept of the phase for second order linear differential equations. Oscillation and nonoscillation of (1) and of symplectic systems are investigated in connection with phases and trigonometric systems. Some applications to summation of number series are given, too.
The concept of characteristic interval for piecewise monotone functions is introduced and used in the study of their iterative roots on a closed interval.
Starting with the computation of the appropriate Poisson kernels, we review, complement, and compare results on drifted Laplace operators in two different contexts: homogeneous trees and the hyperbolic half-plane.