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When the literature is examined, it is seen that there are many studies on the generalizations of gamma, beta and hypergeometric functions. In this paper, new types of generalized gamma and beta functions are defined and examined using the Wright function. With the help of generalized beta function, new type of generalized Gauss and confluent hypergeometric functions are obtained. Furthermore, some properties of these functions such as integral representations, derivative formulas, Mellin transforms, Laplace transforms and transform formulas are determined. As examples, we obtained the solution of fractional differential equations involving the new generalized beta, Gauss hypergeometric and confluent hypergeometric functions. Finally, we presented their relationship with other generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions, which can be found in the literature.

The main purpose of the present paper is to determine the radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B, Comment: arXiv admin note: substantial text overlap with arXiv:1911.07901

Many authors have established various integral and differential formulas involving different special functions in recent years. In continuation, we explore some image formulas associated with the product of Srivastava’s polynomials and extended Wright function by using Marichev–Saigo–Maeda fractional integral and differential operators, Lavoie–Trottier and Oberhettinger integral operators. The obtained outcomes are in the form of the Fox–Wright function. It is worth mentioning that some interesting special cases are also discussed.

In this paper, we introduced the Mainardi beam and indicated its attributes under the Fractional Fourier transform for power variations of Fractional Fourier transform. The results represent that the behavior of the Mainardi beam is similar to that of the Airy beam. The obtained formula is a very powerful tool to describe propagation of a Mainardi beam through the FFT and the FrFT systems. An analytical expression of the Mainardi beam passing through an Fractional Fourier transform system presented. The influences of the Fractional Fourier transform, rational order of the Mittag-Leffler function (Fourier transform of the Mainardi function) on the normalized intensity distribution and characteristics of the Mainardi beam in the Fractional Fourier transform system examined. Power of the Fractional Fourier transform (p) and rational order of the Mittag-Leffler function (q) control characteristics of the Mainardi beam such as effective beam size, number, width, height, and orientation of the beam spot.

In the present paper, we investigate certain geometric properties and inequalities for the Wright function and mention a few important consequences of our main results. A nonlinear differential equation involving the Wright function is also investigated.

The asymptotic expansion for x→±∞ of the entire function Fn,σ(x, μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk),γk=(k+1)π2n for μ&gt, 0, 0&lt, σ&lt, 1 and n=1,2,… is considered. In the special case σ=α/(2n), with 0&lt, α&lt, 1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn,σ(x, μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained.

The aim of this paper to establish the Marichev-Saigo-Maeda fractional integration formula to the product of the K-function with the general class of multivariable polynomials. The results are presented in terms of the Wright generalized hypergeometric function. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals are also presented. Further, we point out also their relevance.

The close form of some integrals involving recently developed generalized k-Struve functions is obtained. The outcome of these integrations is expressed in terms of generalized Wright functions. Several special cases are deduced which lead to some known results.

The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed.

Two Stefan’s problems for the diffusion fractional equation are solved, where the fractional derivative of order α ∈ (0, 1) is taken in the Caputo sense. The first one has a constant condition on x = 0 and the second presents a flux condition Tx(0, t) = q t α/2. An equivalence between these problems is proved and the convergence to the classical solutions is analyzed when α 1 recovering the heat equation with its respective Stefan’s condition. Fil: Roscani, Sabrina Dina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Santillan Marcus, Eduardo Adrian. Universidad Austral. Facultad de Ciencias Empresariales; Argentina

In a previous note a generalized Beer's law was discussed in relation to the space-fractional Poisson process to explain possible deviations from the exponential extinction law in spatially correlated media. Here a different point of view will be developed, applying a Wright type function to describe the probability of photon transmission in random media. We find the analytic form of the photon mean-free-path (MFP) related to such Wright law of extinction. We also give an estimate of the deviation from an exponential law, showing the utility of our approach and discussing a comparison with the predictions given by the classical Beer's law in uniform media.

We propose an extension of the cable equation by introducing a Caputo time fractional derivative. The fundamental solutions of the most common boundary problems are derived analitically via Laplace Transform, and result be written in terms of known special functions. This generalization could be useful to describe anomalous diffusion phenomena with leakage as signal conduction in spiny dendrites. The presented solutions are computed in Matlab and plotted., Comment: 10 figures. arXiv admin note: substantial text overlap with arXiv:1702.05325

We consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function, or Wright function, "[email protected]"q(z) for large |z| that have not been considered in the existing theory. Our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis. Numerical examples are presented to demonstrate the presence of these exponentially small expansions.

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations., 43 pages, 19 figures, in press on Physica A (2008)

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0 1 2 or 1 2 , respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and Signalling problems can be expressed in terms of an auxiliary function M(z;β), where z = |x| t β is the similarity variable. Such function is proved to be an entire function of Wright type.

The purpose of the present paper is to investigate some characterizations for the Wright functions to be in the subclasses S * ( α , β ) and C ( α , β ) ( α , β ∈ [ 0 , 1 ) ) . Several sufficient conditions are obtained for the normalized Wright functions to be in these classes. Results obtained in this paper are new and their usefulness is put forth by several corollaries.

The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of time orders we provide the fundamental solution, that is still a probability density, in terms of an integral of Laplace type. The kernel depends on the type of the assumed fractional derivative except for the single order case where the two approaches turn to be equivalent. We consider with some detail two cases of order distribution: the double-order and the uniformly distributed order. For these cases we exhibit plots of the corresponding fundamental solutions and their variance, pointing out the remarkable difference between the two approaches for small and large times., Comment: 30 pages, 4 figures. International Workshop on Fractional Differentiation and its Applications (FDA06), 19-21 July 2006, Porto, Portugal. Journal of Vibration and Control, in press (2007)

After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding renewal processes with reward and numerically their long-time behaviour., 2000 MSC: 26A33, 33E12, 33E20, 44A10, 44A35, 60G50, 60J05, 60K05.

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 we obtain solutions written in terms of Wright functions by exploiting operational rules involving the shift operator. We also consider fractional advection diffusion equations involving fractional powers of the negative Laplace operator and directional derivatives of fractional order and discuss the probabilistic interpretations of solutions.