Skip to main content

Theory and Modern Applications

Symmetric Three-Term Recurrence Equations and Their Symplectic Structure


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.

Publisher note

To access the full article, please see PDF

Author information

Authors and Affiliations


Corresponding author

Correspondence to Roman Šimon Hilscher.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Šimon Hilscher, R., Zeidan, V. Symmetric Three-Term Recurrence Equations and Their Symplectic Structure. Adv Differ Equ 2010, 626942 (2010).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation