The Mittag-Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

The computation of the Mittag-Leffler (ML) function with matrix arguments, and some applications in fractional calculus, are discussed. In general the evaluation of a scalar function in matrix arguments may require the computation of derivatives of possible high order depending on the matrix spectrum. Regarding the ML function, the numerical computation of its derivatives of arbitrary order is a completely unexplored topic; in this paper we address this issue and three different methods are tailored and investigated. The methods are combined together with an original derivatives balancing technique in order to devise an algorithm capable of providing high accuracy. The conditioning of the evaluation of matrix ML functions is also studied. The numerical experiments presented in the paper show that the proposed algorithm provides high accuracy, very often close to the machine precision. , avialable at https://doi.org/ 10.1007/s10915-018-0699-5. This work is supported under the GNCS-INdAM 2017 project "Analisi numerica per modelli descritti da operatori frazionari". arXiv:1804.04883v2 [math.NA] 1 Dec 2019where |δ 1 |, |δ 2 |, . . . , |δ J | < and terms proportional to O( 2 ) have been discarded. It is thus immediate to derive the following bound for the round-off errorThe order by which the terms c j are summed is relevant especially for the reliability of the above estimator; clearly, an ascending sorting of |c j | makes the estimate (16) more conservative and hence more useful for practical use. It is therefore advisable, especially in the more compelling cases (namely, for arguments z with large modulus and | arg(z)| > απ/2) to perform a preliminary sorting of the terms in (14) with respect to their modulus.An alternative estimate of the round-off error can be obtained after reformulating (15) asŜ J = S J + J j=1 S j δ j from which one can easily derive S J −Ŝ J ≤ J j=1

This paper addresses the problem of the numerical computation of generalized Mittag-Leffler functions with two parameters, with applications in fractional calculus. The inversion of their Laplace transform is an effective tool in this direction; however, the choice of the integration contour is crucial. Here parabolic contours are investigated and combined with quadrature rules for the numerical integration. An in-depth error analysis is carried out to select suitable contour's parameters, depending on the parameters of the Mittag-Leffler function, in order to achieve any fixed accuracy. We present numerical experiments to validate theoretical results and some computational issues are discussed.