AUTHOR=Shat Rasha , Alrefai Safa , Alhamayda Islam , Sarhan Alaa , Al-Refai Mohammed TITLE=The Fractional Laguerre Equation: Series Solutions and Fractional Laguerre Functions JOURNAL=Frontiers in Applied Mathematics and Statistics VOLUME=5 YEAR=2019 URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2019.00011 DOI=10.3389/fams.2019.00011 ISSN=2297-4687 ABSTRACT=

In this paper, we propose a fractional generalization of the well-known Laguerre differential equation. We replace the integer derivative by the conformable 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 solution 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.