Spectral decomposition of the fractional Sturm–Liouville operator with $\rho $-generalized derivative

We study the $\rho $-generalized fractional version of the classical Sturm–Liouville operator. We establish the existence and uniqueness of a boundary value problem for this operator with homogeneous boundary conditions. Using this problem, we present a spectral decomposition of the $\rho $-generalized fractional Sturm–Liouville operator.