New Approaches for Solving Interpolation Problems and Homogeneous Linear Recurrence Relations

 This article presents a new approach to address the resolution of homogeneous linear recurrences of higher order and interpolation problems. By establishing an explicit formula for the entries of the inverse of generalized Vandermonde matrices, a fresh perspective on these mathematical challenges is introduced. The study primarily focuses on linear recurrence relations and thoroughly investigates cases involving characteristic polynomials with both simple roots and roots of multiplicity. To illustrate the effectiveness and practicality of the proposed method, a comprehensive set of illustrative examples is provided, highlighting its applicability in solving a wide range of instances of linear recurrence relations. Additionally, the limitations of the formula are discussed, particularly in scenarios where its applicability may be restricted. The findings of this study contribute significantly to the existing literature, providing an alternative and promising approach for solving problems that rely on the inverse Vandermonde matrix. In conclusion, this article emphasizes the need for further research to explore the computational advantages of the proposed method and to extend its applicability to cases featuring characteristic polynomials with a single root of multiplicity greater than one. By expanding the knowledge in the field, this study offers valuable insights into the resolution of linear recurrences and interpolation problems, presenting a new perspective and expanding the existing knowledge in the field.