International Journal of Advanced Engineering Research and Science(IJAERS) Vol-9, Issue-6; Jun, 2022 INVERSE OF THE GENERALIZED VANDERMONDE MATRIX VIA THE FUNDAMENTAL SYSTEM OF LINEAR DIFFERENCE EQUATIONS CLAUDEMIR ANIZ AND MUSTAPHA RACHIDI Instituto de Matem´ atica INMA, UFMS, Av. Costa e Silva Cidade Universitaria, Campo Grande - MS - Brazil ABSTRACT. In this study we display a process for inverting the generalized Van- dermonde matrix, using the analytic properties of a fundamental system related to a specific linear difference equations. We establish two approaches allowing us to provide explicit formulas for the entries of the inverse of the generalized Vandermonde matrices. To enhance the effectiveness of our the approaches, sig- nificant examples and special cases are given. Key Words: Generalized Vandermonde matrix; Inverse of the generalized Vander- monde matrix; Linear difference equations; Analytic formulas; Fibonacci funda- mental system. 2010 Mathematical Subject Classifications: 11B99; 15B99; 65F05; 65Q10; 97N50. 1. I NTRODUCTION The usual Vandermonde systems of equations of order r is given by, r  i=1 λ n i x i = v n , n =0, 1,...,r − 1, (1) where the x i (1 ≤ i ≤ r) are the unknown variables, the λ i (1 ≤ i ≤ r) are distinct real (or complex) numbers and the v n (0 ≤ n ≤ r − 1) given real (or complex) numbers. Let m i ≥ 1 (1 ≤ i ≤ s) be s integers and λ i (1 ≤ i ≤ s) be distinct real (or complex) numbers. For a given real (or complex) numbers v n (0 ≤ n ≤ r − 1), where r = m 1 + ··· + m s , the related generalized Vandermonde systems of equations is defined as follows, s  i=1   m i −1  j =0 x i,j n j   λ n i = v n , n =0, 1,...,r − 1, (2) where the x i,j (1 ≤ i ≤ s, 0 ≤ j ≤ m i − 1) are the unknown variables. The generalized Vandermonde system of equations is also known in the literature as a nonsingular usual Vandermonde system. The generalized Vandermonde systems (1)-(2) appear in several topics of mathematics such that the linear algebra, numerical analysis and polynomial e-mails addresses: claudemir.aniz@ufms.br, mustapha.rachidi@ufms.br, mu.rachidi@gmail.com. 106