Related papers: Extended Mittag-Leffler Function and truncated $\n…

In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local…

We develop a theory of "special functions" associated to a certain fourth order differential operator $\mathcal{D}_{\mu,\nu}$ on $\mathbb{R}$ depending on two parameters $\mu,\nu$. For integers $\mu,\nu\geq-1$ with $\mu+\nu\in2\mathbb{N}_0$…

We defined and used a pair of Hermitian annihilation and creation operators which generate the generalized coherent states, defined in the Barut-Girardello manner, whose normalization function is just the four-parameter generalized…

There have been many proposed forms of fractional calculus, which can be grouped into a few broad classes of operators. By replacing the kernel of the power function with another kernel function, the traditional Riemann-Liouville formula…

For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for…

In this paper, we introduce the little $\mu$-function, which is obtained as a degenerate limit of the generalized $\mu$-function. We derive the little $\mu$-function as the image of the $q$-Borel summation of a divergent solution to the…

In this paper, we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving…

We review the recent generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives and study them using indirect methods. In particular, we provide necessary…

The Mittag-Leffler function plays an important role in Geometric Function Theory, particularly in the study of analytic and meromorphic function classes. Among its various generalizations, the Barnes-Mittag-Leffler function stands out due…

We first prove semi-orthogonal decompositions of derived factorization categories arising from sums of potentials of gauged Landau-Ginzburg models, where the sums are not necessarily Thom--Sebastiani type. We then apply the result to the…

This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant…

In this paper we study the regularity properties of fractional maximal operators acting on $BV$-functions. We establish new bounds for the derivative of the fractional maximal function, both in the continuous and in the discrete settings.

Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…

It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we…

We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…

A strong inspiration for studying Sobolev type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev type…

A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…

Recently, a new fractional derivative called the conformable fractional derivative is given on based basic limit definition derivative in [4]. Then, the fractional versions of chain rules, exponential functions, Gronwalls inequality,…

In this note we show how a initial value problem for a relaxation process governed by a differential equation of non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a…

We give a functorial characterization of Mittag-Leffler modules and strict Mittag-Leffler modules.