The Integral Mittag-Leffler, Whittaker and Wright Functions

In this paper, we consider a natural convection flow of an incompressible viscous fluid subject to Newtonian heating and constant mass diffusion. The proposed model has been described by the Caputo fractional operator. The used derivative is compatible with physical initial and boundaries conditions. The exact analytical solutions of the proposed model have been provided using the Laplace transform method. The obtained solutions are expressed using some special functions as the Gaussian error function, Mittag–Leffler function, Wright function, and G‐function. The influences of the order of the fractional operator, parameters used in modeling the considered fluid, Nusselt number, and Sherwood number have been analyzed and discussed. The physical interpretations of the influences of the parameters of our fluid model have been presented and analyzed as well. We use the graphical representations of the exact solutions of the model to support the findings of the paper.

Chapter 7 - Exact Analytical Solutions for Fractional Viscoelastic Fluids

Special functions such as hypergeometric, zeta, Bessel, Whittaker, Struve, Airy, Weber-Hermite and Mittag-Leffler functions are obtained as a solution to complex differential equations in engineering, science and technology. In this work, generalized Euler-type integrals involving four parameters Mittag-Leffler function are proposed. Some special cases of this type of generalized integrals that are corresponding to well-known results in the literature are also inferred.

ABSTRACTIn this paper, using the Mellin transform of Wright function we derive an addition formula for the Wright function. In some special cases, addition formulas for the Hermite, Bessel and Mittag-Leffler functions are also given and the Green's function of two-dimensional time-fractional diffusion equation is presented in the whole plane.

We introduce a novel class of coherent states, termed Wα,β,ν-coherent states, constructed using a deformed boson algebra based on the generalised factorial [n]α,β,ν!. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analysed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between the classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies.

ON A NEW CLASS OF FUNCTIONS RELATED WITH MITTAG-LEFFLER AND WRIGHT FUNCTIONS AND THEIR PROPERTIES

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

This paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory. We establish new asymptotic properties of this function under numerous assumptions on the ordering of the parameter and variable. Several representations involving well-known special functions are given. We also provide some integral representations and structural properties involving the “reduced” Wright function. Some of the properties established imply a reflection principle that connects the “reduced” Wright functions with the opposite values of the parameter and certain Bessel functions. Several asymptotic relations for both particular cases of this function are also given.

The formal term-by-term differentiation with respect to parameters is demonstrated to be legitimate for the Mittag–Leffler-type functions. The justification of differentiation formulas is made by using the concept of the uniform convergence. This approach is applied to the Mittag–Leffler function depending on two parameters and, additionally, for the three-parametric Mittag–Leffler functions (namely, for the Prabhakar function and the Le Roy-type function), as well as for the four-parametric Mittag–Leffler function (and, in particular, for the Wright function). The differentiation with respect to the involved parameters is discussed also in case those special functions which are represented via the Mellin–Barnes integrals.

Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996) introduced and investigated certain generalized fractional integral and derivative operators involving the Appell function . Here we aim at presenting four unified fractional integral and derivative formulas of Saigo and Maeda type, which are involved in a product of ℵ-function and a general class of multivariable polynomials. The main results, being of general nature, are shown to be some unification and extension of many known formulas given, for example, by Saigo and Maeda (Transform Methods and Special Functions, Varna, Bulgaria, pp. 386-400, 1996), Saxena et al. (Kuwait J. Sci. Eng. 35(1A):1-20, 2008), Srivastava and Garg (Rev. Roum. Phys. 32:685-692, 1987), Srivastava et al. (J. Math. Anal. Appl.193:373-389, 1995) and so on. Our main results are also shown to be further specialized to yield a large number of known and (presumably) new formulas involving, for instance, Saigo fractional calculus operators, several special functions such as H-function, I-function, Mittag-Leffler function, generalized Wright hypergeometric function, generalized Bessel-Maitland function. MSC:26A33, 33E20, 33C45, 33C60, 33C70.

Amplitudes for scattering of electrons by hydrogenic and alkali-like atomic systems

Recently, some researchers determined lower bounds for the normalized version of some special functions to its sequence of partial sums, e.g., Struve and Dini functions, Wright functions and Miller–Ross functions. In this paper, we determine lower bounds for the normalized Le Roy-type Mittag-Leffler function Fα,βγ(z)=z+∑n=1∞Anzn+1, where An=ΓβΓα(n−1)+βγ and its sequence of partial sums (Fα,βγ(z))m(z)=z+∑n=1mAnzn+1. Several examples of the main results are also considered.

Some new re-lines with special functions for their solution

The single parameter Mittag-Leffler function, which is a generalization of the exponential function, occurs naturally in the solution of physical problems involving fractional calculus and its zeros play a significant role in the dynamic solutions. It is known that the Mittag-Leffler function has a finite number of real zeros in the range of parameter values between 1 and 2, which is quite relevant for many physical problems. However, the number of real zeros for a given parameter value in this range is not known. An iteration formula for calculating the number of real zeros of the Mittag-Leffler function for any value of the parameter in the range between 1 and 2 is presented.

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.