The Mittag-Leffler function
英文摘要：
We review the function theoretical properties of the Mittag-Leffler function $E_{a,b}\left( z\right) $ in a self-contained manner, but also add new results; more than half is new!
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Weighted Composition Operator and the Mittag-Leffler space
We present important characterizations of the Weighted Composition Operator over the Mittag Leffler space of entire functions. These characterizations include the Hilbert-Schmidt and Unitary char-acterizations of the Weighted Composition Operator over the Mittag-Leffler space. We also see the result for the spectrum of Weighted com-position operator over the Mittag-Leffler space
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Mittag-Leffler Analysis II: Application to the fractional heat equation
Mittag-Leffler analysis is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which generalizes the powerful theory of Gaussian analysis and in particular white noise analysis. In this paper we further develop the Mittag-Leffler analysis by characterizing the convergent sequences in the distribution space. Moreover we provide an approximation of Donsker's delta by square integrable functions. Then we apply the structures and techniques from Mittag-Leffler analysis in order to show that a Green's function to the time-fractional heat equation can be con…
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On Generalized Fractional Derivatives Involving Generalized k-Mittag Leffler Function
In this paper, certain generalized fractional derivative formulae are introduced involving the k-Mittag-Leffler function. Then their image formulae (using Beta transform, Laplace transform and Whittaker transform) are also established. The results obtained here are quite general in nature. The special cases of our findings are also discussed.
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Mittag-Leffler Analysis I: Construction and characterization
We construct an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak inte…
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非高斯 Mittag-Leffler 泛函上算子符号的解析刻画
An analytic characterization of symbols of operators on non-Gaussian Mittag-Leffler functionals
该研究借助 Mittag-Leffler 分布空间刻画定理，建立非高斯 Mittag-Leffler 泛函上算子符号的解析刻画，并通过积分核算子与平移算子示例说明。
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The Mittag Leffler space & its Littlewood-Paley Identity
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关于 Mittag-Leffler 函数的 van der Corput 引理。I
Van der Corput lemmas for Mittag-Leffler functions. I
本文将经典van der Corput引理推广至Mittag-Leffler型函数，用于研究时间分数阶偏微分方程中的振荡型积分，证明了第一、第二引理的推广及衰减阶最优估计，并应用于广义Riemann-Lebesgue引理与时间分数阶Schrödinger方程的Cauchy问题。
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Van der Corput lemmas for Mittag-Leffler functions. II. $α$-directions
The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form $I_{α,β}(λ)=\int_\mathbb{R}E_{α,β}\left(i^αλϕ(x)\right)ψ(x)dx,$ for the range $0<α\leq 2,\,β>0$. This extends the variety of estimates obtained in the first part, where integrals with functions $E_{α,β}\left(i λϕ(x)\right)$ have been st…
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Integral representations and asymptotic behaviours of Mittag-Leffler type functions of two variables
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Uniqueness and Ulam-Hyers-Mittag-Leffler stability results for the delayed fractional multi-terms differential equation involving the $Φ$--Caputo fractional Derivative
The principal aim of the present paper is to establish the uniqueness and Ulam-Hyers Mittag-Leffler (UHML) stability of solutions for a new class of multi-terms fractional time-delay differential equations in the context of the $Φ$-Caputo fractional derivative. To achieve this purpose, the generalized Laplace transform method alongside facet with properties of the Mittag-Leffler functions (M-LFs), are utilized to give a new representation formula of the solutions for the aforementioned problem. Besides that, the uniqueness of the solutions of the considered problem is also proved by applying t…
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Integral representations and asymptotic behaviour of a Mittag-Leffler type function of two variables
Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases.The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and…
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Fractional calculus and generalized Mittag-Leffler type functions
In this paper, the generalized fractional integral operators of two generalized Mittag-Leffler type functions are investigated. The special cases of interest involve the generalized Fox--Wright function and the generalized M-series and K-function. In the next Section 2 we first recall some generalized fractional integral operators among the most widely used in fractional calculus. Section 3 is devoted to the definitions of M-series and K-function and their relations to special functions. In Sections 4 and 5, effective fractional calculus of the generalized M-series and the K-function is carrie…
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