Construction and Orthogonality of Fractional Laguerre Functions via the Caputo Derivative

Abstract

This paper presents a rigorous framework for generalizing Laguerre polynomials to the fractional domain using the Caputo derivative. We solve the resulting fractional Laguerre differential equation via the power series method, deriving an explicit formfor the fractional Laguerre functions. A key contribution is the identification of a novel weight function, wα(x) = x−(2α−1)e−x, which is essential to prove the orthogonality of these functions over the interval [0,∞). Comprehensive numerical validation is provided, confirming the theoretical orthogonality across a wide range of fractional orders α and demonstrating a clean reduction to the classical polynomials when α = 2. An analysis of computational feasibility confirms the practical applicability of these functions for solving fractional differential equations and other applied problems.

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