The Ambartsumian equation is studied and solved. A simple solution expressed in terms of a linear combination of Mittag-Leffler functions is deduced and implemented. A fast algorithm for its numerical realization is described and examplified.

The substantial, tempered, and shifted fractional derivatives are introduced in a unified framework. Their properties are studied and, in the light of the strict sense criterion for derivative definitions, they are characterized and assessed. In the scope of the framework, new tempered linear sytems and transfer functions are introduced.

We construct an operational calculus for a modified shift operator on an abstract space of formal Laurent series. We show that this calculus is a universal model for a large number of operational calculi for several kinds of differential and difference operators.