2025, Vol. 6, Issue 1, Part D

Legendre polynomials form a fundamental class of classical orthogonal polynomials that play a pivotal role in the development and application of orthogonal function systems. Defined as solutions to Legendre’s differential equation, these polynomials possess key properties such as orthogonality, completeness, and well-structured recurrence relations on the interval [−1,1]. The present study examines the theoretical foundations of Legendre polynomials, situating them within the broader framework of orthogonal function systems and Sturm-Liouville theory. Emphasis is placed on their analytical characteristics, including generating functions, normalization, and convergence behavior of Legendre series. The paper also highlights the significance of Legendre polynomials in approximation theory and their effectiveness in solving boundary value problems and partial differential equations. Furthermore, attention is given to their extensions and comparative relevance alongside other classical orthogonal systems. Overall, the study underscores the continued importance of Legendre polynomials as both a theoretical construct and a practical tool in modern mathematical analysis and applied sciences.