Fractional Laguerre derivatives and associated differential equations

Abstract

Fractional versions of the Laguerre derivative were introduced in 2021 by Yakubovich, who defined both fractional integrals and fractional derivatives (of Riemann-Liouville type) within the Laguerre setting. We continue with the mathematical study of these operators: proving many equivalent formulae for them, introducing the corresponding Caputo-type Laguerre fractional derivatives, and finding eigenfunctions for the new operators. Applications of our work are given by solving several fractional differential equations involving Laguerre-type operators, their solutions often being expressible in terms of special functions such as the multi-index Mittag-Leffler function.