Finite Fractional Sturm–Liouville Transforms For Generalized Fractional Derivatives

In this article, we introduce the regular and singular fractional Sturm–Liouville problems with the Hilfer and Hilfer–Prabhakar derivatives. We show that these problems with the corresponding boundary conditions have real eigenvalues and their eigenfunctions are orthogonal. Also, the finite fractional Sturm–Liouville transforms and their inversion formulas are established, and as an application, the formal solution of the fractional Laplace equation in prolate spheroidal coordinates is obtained using the finite Legendre transform.