Fox Wright Function Research Articles

Generalized Mittag-Leffler-confluent hypergeometric functions in fractional calculus integral operator with numerical solutions

The Mittag-Leffler and confluent hypergeometric functions were originally developed to extend the exponential function and its area of applications. This study aims to examine some operators involving generalized Mittag-Leffler-type functions in the kernels, employing the generalized Fox-Wright function in specific circumstances. Furthermore, we investigate some of the commonly utilized generalized fractional integral operators in fractional calculus. Moreover, a numerical technique is developed to solve fractional differential equations of both kinds, linear and nonlinear. The graphic results of the examples show how effective this method is at solving fractional differential equations. Lastly, various effects and implications of these results are thoroughly examined.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

The Generalized Fox–Wright Function: The Laplace Transform, the Erdélyi–Kober Fractional Integral and Its Role in Fractional Calculus

In this paper, we consider and study in detail the generalized Fox–Wright function Ψ˜qp introduced in our recent work as an extension of the Fox–Wright function Ψqp. This special function can be seen as an important case of the so-called I-functions of Rathie and H¯-functions of Inayat-Hussain, that in turn extend the Fox H-functions and appear to include some Feynman integrals in statistical physics, in polylogarithms, in Riemann Zeta-type functions and in other important mathematical functions. Depending on the parameters, Ψ˜qp is an entire function or is analytic in an open disc with a final radius. We derive its basic properties, such as its order and type, and its images under the Laplace transform and under classical fractional-order integrals. Particular cases of Ψ˜qp are specified, including the Mittag-Leffler and Le Roy-type functions and their multi-index analogues and many other special functions of Fractional Calculus. The corresponding results are illustrated. Finally, we emphasize the role of these new generalized hypergeometric functions as eigenfunctions of operators of new Fractional Calculus with specific I-functions as singular kernels. This paper can be considered as a natural supplement to our previous surveys “Going Next after ‘A Guide to Special Functions in Fractional Calculus’: A Discussion Survey”, and “A Guide to Special Functions of Fractional Calculus”, published recently in this journal.

Uncertain Asymptotic Stability Analysis of a Fractional-Order System with Numerical Aspects

We apply known special functions from the literature (and these include the Fox H–function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the G–function, the Fox–Wright function and the Meijer G–function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory.

FRACTIONAL CALCULUS AND FRACTIONAL KINETIC EQUATION INVOLVING EXTENDED HYPERGEOMETRIC FUNCTION

In this article, generalized pathway integral operator with the classical Gauss hypergeometric function kernel and the fractional differential operators are used to studied new extended hypergeometric function. Furthermore, extended fractional kinetic equation involving new extended hypergeometric function that contained two Fox-Wright function as regularizer is investigated using Sumudu transform and Wiman’s function

Distributional Representation of a Special Fox–Wright Function with an Application

A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, we studied a new representation of this function in terms of complex delta functions. This representation was useful to compute its Laplace transform with respect to the third parameter γ for which it also generalizes the one and two-parameter Mittag-Leffler functions. New identities involving the Fox–Wright function were discussed and used to simplify the results. Different fractional transforms were evaluated and the solution of a fractional kinetic equation was obtained by using its new representation. Several new properties of this function were discussed as a distribution.

A new Euler-Beta function model with statistical implementation related to the Mittag-Leffler-Kummer function

Recently, the realm related to Euler's Beta function has played a significant role in the development of special function theory. In this study, a new extension of the special function known as Euler's Beta function with respect to the Mittag-Leffler-Kummer function is introduced. Another formula of this new Beta function in terms of the Fox-Wright function is also presented. Numerous analytical properties of this new special function, such as several functional and summation relations, Mellin transforms, integral representations, and several derivative formulas, are studied. Furthermore, significant statistical implementations of Euler's Beta distribution are also discussed.

On Novel Fractional Integral and Differential Operators and Their Properties

The main goal of this paper is to describe the new version of extended Bessel–Maitland function and discuss its special cases. Then, using the aforementioned function as their kernels, we develop the generalized fractional integral and differential operators. The convergence and boundedness of the newly operators and compare them with the existing operators such as the Saigo and Riemann–Liouville fractional operators are explored. The integral transforms of newly defined and generalized fractional operators in terms of the generalized Fox–Wright function are presented. Additionally, we discuss a few exceptional cases of the main result.

New Results Involving the Generalized Krätzel Function with Application to the Fractional Kinetic Equations

Sun is a basic component of the natural environment and kinetic equations are important mathematical models to assess the rate of change of chemical composition of a star such as the sun. In this article, a new fractional kinetic equation is formulated and solved using generalized Krätzel integrals because the nuclear reaction rate in astrophysics is represented in terms of these integrals. Furthermore, new identities involving Fox–Wright function are discussed and used to simplify the results. We compute new fractional calculus formulae involving the Krätzel function by using Kiryakova’s fractional integral and derivative operators which led to several new identities for a variety of other classic fractional transforms. A number of new identities for the generalized Krätzel function are then analyzed in relation to the H-function. The closed form of such results is also expressible in terms of Mittag-Leffler function. Distributional representation of Krätzel function and its Laplace transform has been essential in achieving the goals of this work.

COMPLEX MATHEMATICAL MODELING FOR ADVANCED FRACTAL–FRACTIONAL DIFFERENTIAL OPERATORS WITHIN SYMMETRY

Non-local operators of differentiation are bestowed with capabilities of encompassing complex natural into mathematical equations. Symmetry as invariance under a specified group of transformations can allow for the concept to be applied extensively not only to spatial figures but also to abstract objects like mathematical expressions which can be said to be expressions of physical relevance, in particular dynamical equations. Derived from this point of view, it can be noted that the more complex physical problems are, the more complex mathematical operators of differentiation are required. Accordingly, the fractal–fractional operators (FFOs) are expanded into the complex plane in our research which revolves around a unique class of normalized analytic functions in the open unit disk. To bring FFOs (differential and integral) into the normalized class, the study aims to expand and modify them along with the investigation of the FFOs geometrically. The qualities of convexity and starlikeness are implicated in this study where the differential subordination technique serves as the foundation for the inquiry under consideration. Furthermore, a collection of differential FFO inequalities is taken into account, demonstrating that the normalized Fox–Wright function can contain all FFOs. Besides these steps, the concept of Grunsky factors is applied to investigate symmetry, while boundary value issues involving FFOs are probed. Consequently, the related properties and applications can be further developed, which requires the devotion to differential fractional problems and diverse complex problems in relation to viable applications, pointing out the room to modify and upgrade the existing methods for more optimal outcomes in challenging real-world problems.

On the Generalization of Fractional Kinetic Equation Comprising Incomplete H-Function

In the present work, a novel and even more generalized fractional kinetic equation has been formulated in terms of polynomial weighted incomplete H-function, incomplete Fox-Wright function and incomplete generalized hypergeometric function, considering the importance of the fractional kinetic equations arising in the various science and engineering problems. All the derived findings are of natural type and can produce a variety of fractional kinetic equations and their solutions.

CERTAIN GENERALIZED FRACTIONAL CALCULUS FORMULAS AND INTEGRAL TRANSFORMS OF (p, q)-EXTENDED τ -HYPERGEOMETRIC FUNCTION

In this paper, we established certain image formulas of the (p, q)–extended τ hypergeometric function Rτ p ,q (a, b; c; z) by employing Marichev-Sigo-Maeda(M-S-M) fractional integration and differentation Corresponding special cases for the Saigo’s, Riemann-Liouville and Erdelyi-Kober fractional integral and differ- ential operators are also deduced which are earlier obtained by Solanki et al. [23]. Further certain integral transforms of the (p, q)–extended τ hypergeometric function Rτ (a, b; c; z) are established. All the results are represented in terms of the Hadamard product of the (p, q)–extended τ hypergeometric function Rτ (a, b; c; z) and the Fox-Wright function.

Unified Approach to Fractional Calculus Images Involving the Pathway Transform of Extended k -Gamma Function and Applications

A path way model is concerned with the rules of swapping among different classes of functions. Such model is significant to fit a parametric class of distributions for new data. Based on a such model, the path way or P δ transform is of binomial type and contains a family of different transforms. Taking inspiration from these facts, present research is concerned with the computation of new fractional calculus images involving the extended k -gamma function. The non-integer kinetic equations containing the extended k -gamma function is solved by using pathway transform as well as validated with the earlier obtained results. P δ transform of Dirac delta function is obtained which proved useful to achieve the purpose. As customary, the results for the frequently used Laplace transform can be recovered by taking δ ⟶ 1 in the definition of P δ transform. Important new identities involving the Fox-Wright function are obtained and used to simplify the results. It is remarkable that the several new and novel results involving the classical gamma function became possible by using this approach.

Observations on the McKay Iν Bessel distribution

The main aim of this article is to derive new representation formulae for CDF of random variable distributed according to the McKay Iν Bessel law: first set of formulae are given in terms of incomplete generalized Fox–Wright function while the other expressions include Exton generalized hypergeometric X function of two variables, and also the incomplete Lipschitz–Hankel integral for the modified Bessel function of the first kind. Also we establish the connection between CDF of the McKay Iν Bessel rv and another important CDF which belongs to the non–central chi–square law. Another goal is to give answers to the moment determinacy problem, unimodality and some mode results for this distribution.

New Results Involving Riemann Zeta Function Using Its Distributional Representation

The relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag–Leffler function, became the queen of fractional calculus because its image under the Laplace transform is known to a large audience only in this century. By taking motivation from these facts, we use distributional representation of the Riemann zeta function to compute its Laplace transform, which has played a fundamental role in applying the operators of generalized fractional calculus to this well-studied function. Hence, similar new images under various other popular fractional transforms can be obtained as special cases. A new fractional kinetic equation involving the Riemann zeta function is formulated and solved. Thereafter, a new relation involving the Laplace transform of the Riemann zeta function and the Fox–Wright function is explored, which proved to significantly simplify the results. Various new distributional properties are also derived.

Integral representation and computational properties of the incomplete Fox–Wright function

Integral representation and computational properties of the incomplete Fox–Wright function

Certain Integral and Differential Formulas Involving the Product of Srivastava’s Polynomials and Extended Wright Function

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.

A catalogue of semigroup properties for integral operators with Fox–Wright kernel functions

AbstractThe Fox–Wright function is a very general form of function, covering many families of special functions as particular cases. Any special function can be used as a kernel for a fractional integral operator, but which of these operators will satisfy desiderata such as a semigroup property for composition? This paper provides a rigorous categorisation of all such fractional integral operators which have a semigroup property in any of their parameters. We discover that nearly all possible semigroup properties arise from the Chu–Vandermonde identity, with the Prabhakar fractional calculus emerging as one special case. For any integral operator with such a semigroup property, it is possible to construct a complete model of fractional calculus, including both integral and derivative operators which interact with each other in a natural way.

The Marichev-Saigo-Maeda Fractional-Calculus Operators Involving the (p,q)-Extended Bessel and Bessel-Wright Functions

The goal of this article is to establish several new formulas and new results related to the Marichev-Saigo-Maeda fractional integral and fractional derivative operators which are applied on the (p,q)-extended Bessel function. The results are expressed as the Hadamard product of the (p,q)-extended Gauss hypergeometric function Fp,q and the Fox-Wright function rΨs(z). Some special cases of our main results are considered. Furthermore, the (p,q)-extended Bessel-Wright function is introduced. Finally, a variety of formulas for the Marichev-Saigo-Maeda fractional integral and derivative operators involving the (p,q)-extended Bessel-Wright function is established.

On generalized Bessel\u2013Maitland function

An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by mathcal{J}_{nu , lambda }^{mu }, where mu >0 and lambda ,nu in mathbb{C } get increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of mathcal{J}_{nu ,lambda }^{mu } by applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, the mth derivative of mathcal{J}_{nu ,lambda }^{mu } is considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions mathcal{I}_{nu ,lambda }^{mu } defined by mathcal{I}_{nu ,lambda }^{mu }(z)=i^{-2lambda -nu }mathcal{J}_{ nu ,lambda }^{mu }(iz) of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for mathcal{I}_{nu ,lambda }^{mu }(z) are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.