On general tempered fractional calculus with Luchko kernels

In this paper, we construct the $n$-fold $\psi$-fractional integrals and derivatives and study their properties. This construction is purely based on the approach proposed by Luchko (2021). The fundamental theorems of fractional calculus are formulated and proved for the proposed $n$-fold $\psi$-fractional integrals and derivatives. On the other hand, a suitable generalization of the Luchko condition is presented to discuss the $\psi$-tempered fractional calculus of arbitrary order. We introduce an important class of kernels that satisfy this condition. For the $\psi$-tempered fractional integrals and derivatives of arbitrary order, two fundamental theorems are proven, along with a relation between Riemann–Liouville and Caputo derivatives. Finally, Cauchy problems for the fractional differential equations with the $\psi$-tempered fractional derivatives are solved.