Some Properties Of The Special Function Of Legendre Within Framework Of Fractional Calculus
DOI:
https://doi.org/10.65405/y26ge219Keywords:
Fractional Calculus, Fractional Legendre Functions, Riemann–Liouville Derivative, Rodrigues' Formula, Gamma Function, Recurrence Relations, Fractional Sturm–Liouville ProblemsAbstract
This study aims to extend classical Legendre polynomials to the framework of fractional calculus by introducing a new family of fractional Legendre functions based on the left-sided Riemann–Liouville fractional derivative with the initial point −1, using Rodrigues' formula. The study investigates the analytical and mathematical properties of the proposed functions, including their differential, integral, and recurrence relations, as well as their behavior under varying fractional derivative orders, it also examines the relationship between the proposed family and classical Legendre polynomials, particularly the recovery of classical results when the fractional derivative order approaches integer values. Furthermore, the study highlights the role of the Gamma function in extending mathematical definitions to fractional orders and investigates related differential equations. The findings indicate that orthogonality and eigenfunction properties in fractional Sturm–Liouville problems require additional conditions involving the fractional operator, weight function, and boundary conditions. The proposed functions may provide a foundation for developing analytical and numerical solutions to fractional differential equations, particularly in modeling phenomena characterized by memory effects, nonlocality, and energy dissipation.
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