**3. Applications**

As we have already discussed, the various Mittag–Leffler functions are interesting from the point of view of pure mathematical analysis and fractional calculus [34–37] (see also the correction [16,36,38]). However, it is also important to discuss the motivation for studying these functions from the point of view of real-world applications in science and engineering.

The one-parameter Mittag–Leffler function has already discovered many applications via the AB model, and also previously in relaxation models which involve interpolation between exponential and power-law behaviours [39]. In recent years, the two-parameter and three-parameter Mittag–Leffler functions have also been emerging from real experimental data.

A group of biologists and engineers in Cambridge and London have been experimenting with models for cells and tissues, and discovered that their data fit most closely to an operator involving two-parameter Mittag–Leffler functions [40].

The three-parameter Mittag–Leffler function, sometimes called the Prabhakar function, is closely connected with the phenomenon of Havriliak–Negami relaxation [41], and this has been studied also in the context of fractional relaxation [26,42].

In view of these manifold applications of the Mittag–Leffler functions of one, two and three parameters, we believe that our results herein may also discover applications. The one-parameter Mittag–Leffler function is much more elementary and easier to handle than the two- and three-parameter Mittag–Leffler functions. Thus, reducing the latter to the former should mark a major step forward. The physical processes which are modelled using two- and three-parameter Mittag–Leffler functions may now be more easily analysed using only the one-parameter Mittag–Leffler function.

In particular, we note that numerical computation of Mittag–Leffler functions has been a challenging problem for researchers in recent years [43–45]. Naturally, the one-parameter Mittag–Leffler function is the most straightforward to handle. If we can use relations such as those proved in this paper to express the more advanced Mittag–Leffler functions purely in terms of the most basic one, then it may enable much easier computation of the two- and three-parameter Mittag–Leffler functions than before.
