The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions - Shat, Rasha; Alrefai, Safa; Alhamayda, Islam; Sarhan, Alaa; Al-Refai, Mohammed (Frontiers in Applied Mathematics and Statistics, 2019)
In this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We replace the integer derivative by the conformable fractional derivative of order $0 < α<1.$ We then apply the Frobenius method with the fractional power series expansion to obtain two linearly independent solutions of the problem. For certain eigenvalues, the infinite series solutions truncate to obtain the singular and non-singular fractional Laguerre functions. We obtain the fractional Laguerre functions in closed forms, and establish their orthogonality result. The applicability of the new fractional Laguerre functions is illustrated.