Related papers: On some properties of the Mittag-Leffler function …
We revisit the Mittag-Leffler functions of a real variable $t$, with one, two and three order-parameters $\{\alpha, \beta, \gamma\}$, as far as their Laplace transform pairs and complete monotonicty properties are concerned. These…
For the two-parameter Mittag-Leffler function $E_{\alpha,\beta}$ with $\alpha > 0$ and $\beta \ge 0,$ we consider the question whether $|E_{\alpha,\beta}(z)|$ and $E_{\alpha,\beta}(\Re z)$ are comparable on the whole complex plane. We show…
We consider two operations on the Mittag-Leffler function which cancel the exponential term in the expansion at infinity, and generate a completely monotonic function. The first one is the action of a certain differential-difference…
Hyperbolic complete monotonicity property ($\mathrm{HCM}$) is a way to check if a distribution is a generalized gamma ($\mathrm{GGC}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions…
The monotonicity of the Mittag-Leffler function $E_{\alpha}$ with respect to the parameter $\alpha$ is investigated, via some convex ordering properties for related random variables. In particular, it is shown that the mapping…
This paper is devoted to the study of the $M$-Wright function ($M_{\alpha}(t)$) which is the inverse Laplace transform of the single-parameter Mittag-Leffler (ML) function ($E_{\alpha}(-s)$). Because $E_{\alpha}(-s)$ can be viewed as the…
The two-parameter Mittag-Leffler function $E_{\alpha, \beta}$ is of fundamental importance in fractional calculus. It appears frequently in the solutions of fractional differential and integral equations. Nonetheless, this vital function is…
The function $t \mapsto E_{\alpha}(\lambda t^\alpha)$ is widely regarded as the fractional analogue of the exponential function, yet its algebraic properties remain poorly understood. In particular, standard references lack a rigorous proof…
Using the Bernstein theorem we give a simple proof of the complete monotonicity of the three parameter generalized Mittag-Leffler function $E_{\alpha, \beta}^{\gamma}(-x)$ for $x \geq 0$ and suitably adjusted parameters $\alpha$, $\beta$…
It is noticed that a certain transform of the Mittag-Leffler function Ea is completely monotone for a in [1,2]. Using the explicit expressions of its Bernstein density, an identity in law between suprema of completely asymmetric Levy…
The two-parametric Mittag-Leffler function (MLF), $E_{\alpha,\beta}$, is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical…
The properties of Mittag-Leffler function is reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the…
This paper proposes a global Pad\'{e} approximation of the generalized Mittag-Leffler function $E_{\alpha,\beta}(-x)$ with $x\in[0,+\infty)$. This uniform approximation can account for both the Taylor series for small arguments and…
The introduction of a fractional differential operator defined in terms of the Riemann-Liouville derivative makes it possible to generalize the kinetic equations used to model relaxation in dielectrics. In this context such fractional…
A key characteristic of the anomalous sub-solution equation is that the solution exhibits algebraic decay rate over long time intervals, which is often refered to the Mittag-Leffler type stability. For a class of power nonlinear…
The three parameters Mittag--Leffler function (often referred as the Prabhakar function) has important applications, mainly in physics of dielectrics, in describing anomalous relaxation of non--Debye type. This paper concerns with the…
In this paper, we prove the strict monotonicity in the parameter $\rho$ of the Mittag-Leffler functions $E_\rho(-t^\rho)$ and $t^{\rho -1}E_{\rho,\rho}(-t^\rho)$. Then, these results are applied to solve the inverse problem of determining…
Pollard used contour integration to show that the Mittag-Leffler function is the Laplace transform of a positive function, thereby proving that it is completely monotone. He also cited personal communication by Feller of a discovery of the…
Our aim in this report is to investigate the asymptotic behavior of Mittag-Leffler functions. We give some estimates involving the Mittag-Leffler functions and their derivatives.
This paper uses convolutions of the gamma density and the one-sided stable density to construct higher level densities. The approach is applied to constructing a 4-parameter Mittag-Leffler density, whose Laplace transform is a corresponding…