Efficient Analytical Solutions for Linear and Nonlinear Difference Equations Using the Discrete Putzer Algorithm

The discrete Putzer algorithm is examined in this work as an effective technique for resolving linear and nonlinear difference equations. The approach simplifies calculations and offers analytical answers by utilizing matrix theory and eigenvalue analysis, particularly for higher-order and non-homogeneous systems. Its efficacy is demonstrated by examples such as Fibonacci sequences and population dynamics. The Cayley-Hamilton theorem is applied to increase the algorithm's usefulness and make it a potent tool for dynamic systems in both theoretical and practical settings.