Advances in Difference Equations
Volume 2010 (2010), Article ID 626942, 17 pages
Research Article

Symmetric Three-Term Recurrence Equations and Their Symplectic Structure

1Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
2Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA

Received 11 March 2010; Accepted 1 May 2010

Academic Editor: Martin Bohner

Copyright © 2010 Roman Šimon Hilscher and Vera Zeidan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We revive the study of the symmetric three-term recurrence equations. Our main result shows that these equations have a natural symplectic structure, that is, every symmetric three-term recurrence equation is a special discrete symplectic system. The assumptions on the coefficients in this paper are weaker and more natural than those in the current literature. In addition, our result implies that symmetric three-term recurrence equations are completely equivalent with Jacobi difference equations arising in the discrete calculus of variations. Presented applications of this study include the Riccati equation and inequality, detailed Sturmian separation and comparison theorems, and the eigenvalue theory for these three-term recurrence and Jacobi equations.