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Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in…
It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling…
Derivatives with respect to the parameters of the integral Mittag-Leffler function and the integral Wright function, recently introduced by us, are calculated. These derivatives can be expressed in the form of infinite sums of quotients of…
In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the…
Mass transport problems are ubiquitous in diverse fields of physics and engineering. With the development of fractional calculus, many have taken to studying problems of fractional mass transport either through numerical simulations or…
In this paper, we first discuss the convolution series that are generated by the Sonine kernels from a class of functions continuous on the real positive semi-axis that have an integrable singularity of power function type at the point…
Most of the special functions of mathematical physics are connected with the representation of Lie groups. The action of elements $D$ of the associated Lie algebras as linear differential operators gives relations among the functions in a…
Generalisations of the classical Euler formula to the setting of fractional calculus are discussed. Compound interest and fractional compound interest serve as motivation. Connections to fractional master equations are highlighted. An…
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…
We consider fractional directional derivatives and establish some connection with stable densities. Solutions to advection equations involving fractional directional derivatives are presented and some properties investigated. In particular…
The Wright function, which arises in the theory of the space-time fractional diffusion equation, is an interesting mathematical object which has diverse connections with other special and elementary functions. The Wright function provides a…
In this note we prove some new results about the application of Wright functions of the first kind to solve fractional differential equations with variable coefficients. Then, we consider some applications of these results in order to…
In this paper we introduce a new fractional derivative with respect to another function the so-called $\psi$-Hilfer fractional derivative. We discuss some properties and important results of the fractional calculus. In this sense, we…
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…
Probability distribution theory helps in studying the impact of various dimensions in life while the Mittag-Leffler function and bicomplex are used in electromagnetism, quantum mechanics, and signal theory. Considering the importance of…
The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a…
We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic…
This article is devoted to derivation of the Laplace transforms of the derivatives with respect to parameters of certain special functions, namely, the Mittag-Leffler type, Wright and Le Roy type functions. These formulas show…
We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…
The main objective of the present paper is to introduce and study the function $_pR_q(A, B; z)$ with matrix parameters and investigate the convergence of this matrix function. The contiguous matrix function relations, differential formulas…