Open Access

Symmetric Three-Term Recurrence Equations and Their Symplectic Structure

Advances in Difference Equations20102010:626942

Received: 11 March 2010

Accepted: 1 May 2010

Published: 2 June 2010


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.

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Authors’ Affiliations

Department of Mathematics and Statistics, Faculty of Science, Masaryk University
Department of Mathematics, Michigan State University


© R. Šimon Hilscher and V. Zeidan. 2010

This article is published under license to BioMed Central Ltd. 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.