Hypergeometric Matrix Research Articles

κ-FRACTIONAL DERIVATIVE OPERATOR AND SOLUTION OF FRACTIONAL KINETIC EQUATIONS INVOLVING (κ,𝔍)-SECOND APPELL HYPERGEOMETRIC MATRIX FUNCTIONS WITH THE TWO VARIABLES

In this study, we present a new definition of [Formula: see text])-second Appell hypergeometric matrix functions ([Formula: see text])-SAHMFs). Then, we investigate analytical properties related to the novel matrix function such as derivative formulas and integral representations, and the [Formula: see text]-fractional derivative operators. Additionally, we obtain bilinear and linear generating relations for the [Formula: see text])-SAHMFs. This work presents novel results regarding [Formula: see text])-SAHMFs to solve the fractional kinetic equations.

On the matrix versions of the k analog of ℑ‐incomplete Gauss hypergeometric functions and associated fractional calculus

In this paper, our aim is to introduce a new definition of ( ‐incomplete Wright hypergeometric matrix functions ( ‐IWHMFs) using the ‐incomplete Pochhammer matrix symbol. First, we define the ‐incomplete gamma matrix function and introduce the ‐incomplete Pochhammer matrix symbols. Furthermore, we present differential formulas and integral representation related to these ‐IWHMFs. We have also obtained some results regarding the ‐fractional calculus operators of these ‐IWHMFs. Finally, we investigate the solutions of fractional kinetic equations (FKEs) involving the ‐IWHMFs.

Starlike and Convex functions associated with Hypergeometric Matrix functions

Starlike and Convex functions associated with Hypergeometric Matrix functions

A study of Horn matrix functions and Horn confluent matrix functions

Abstract In this paper, we give the matrix version of Horn’s hypergeometric function and their confluent cases. We also discuss the regions of convergence, system of matrix differential equations of bilateral type, differential formulae and infinite summation formulae satisfied by these hypergeometric matrix functions. With the study of these 23 matrix functions, matrix generalization of Horn’s list of 34 hypergeometric series will be completed.

Some properties of Ψ-gamma, Ψ-beta and Ψ-hypergeometric matrix functions

Abstract In this paper, we investigate the matrix analogues of the Ψ-beta and Ψ-gamma functions, as well as their properties. With the help of the Ψ-beta matrix function (BMF), we introduce the Ψ-Gauss hypergeometric matrix function (GHMF) and the Ψ-Kummer hypergeometric matrix function (KHMF) and derive certain properties for these matrix functions. Finally, the Ψ-Appell and the Ψ-Lauricella matrix functions are defined by applications of the Ψ-BMF, and their integral representations are also given.

On the two variables κ-Appell hypergeometric matrix functions

This work aims to introduce -Appell hypergeometric matrix functions and , where . New relations for these functions, such as the integral representation, important transformation formulas, infinite matrix summation and reduction formulas, are also presented. Finally, new results of the differential formulas associated with -Appell hypergeometric matrix functions are derived.

Some Relations on the rRs(P,Q,z) Matrix Function

In this paper, we derive some classical and fractional properties of the rRs matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials.

Euler-type integrals for the generalized hypergeometric matrix function

Abstract The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 {|z|=1} . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.

The incomplete matrix beta function and its application to first Appell hypergeometric matrix function

In this paper, we first introduce the incomplete beta matrix function B z ( M , K ) , z ∈ C , for parameter matrices M and K in C m × m using the neutrix limit and evaluation methods on their the partial derivatives. Also, we define a new extension of the Pochhammer matrix symbol in terms of the incomplete beta matrix functions. Furthermore, we then apply this extended matrix symbol for the incomplete first Appell hypergeometric matrix functions and obtain some of their properties.

A novel Beta matrix function via Wiman matrix function and their applications

Abstract Many authors defined and extended the beta function in various forms because the beta function has wide uses in different fields of science and applied science. In this article, we define a new more generalized form of the extended beta matrix function via the Wiman matrix function and describe their significant properties and special cases. Furthermore, we define an extension of the Gauss hypergeometric and confluent hypergeometric matrix functions by adopting a novel type of beta matrix function. We also derive their Laplace transform, derivative formula and transformation formulae.

Some Properties of Incomplete First Appell Hypergeometric Matrix Functions

Some Properties of Incomplete First Appell Hypergeometric Matrix Functions

Investigation of Fractional Calculus for Extended Wright Hypergeometric Matrix Functions

Throughout this paper, we will present a new extension of the Wright hypergeometric matrix function by employing the extended Pochhammer matrix symbol. First, we present the extended hypergeometric matrix function and express certain integral equations and differential formulae concerning it. We also present the Mellin matrix transform of the extended Wright hypergeometric matrix function. After that, we present some fractional calculus findings for these expanded Wright hypergeometric matrix functions. Lastly, we present several theorems of the extended Wright hypergeometric matrix function in fractional Kinetic equations.

On the extended Wright hypergeometric matrix function and its properties

Recently, Bakhet et al. [9] presented the Wright hypergeometric matrix function $_{2}R_{1}^{(\tau )}(A,B;C;z)$ and derived several properties. Abdalla [6] has since applied fractional operators to this function. In this paper, with the help of the generalized Pochhammer matrix symbol $(A;B)_{n}$ and the generalized beta matrix function $\mathcal{B}(P,Q;\mathbb{X})$, we introduce and study an extended form of the Wright hypergeometric matrix function, $_{2}R_{1}^{(\tau )}((A,\mathbb{A}),B;C;z;\mathbb{X}).$ We establish several potentially useful results for this extended form, such as integral representations and fractional derivatives. We also derive some properties of the corresponding incomplete extended Wright hypergeometric matrix function.

Investigation for the k-analogue of τ-Gauss hypergeometric matrix functions and associated fractional calculus

In this manuscript, we present a new definition of ( k , τ ) -Gauss hypergeometric matrix function and study its analytical properties, like derivative formulas and integral representations. Furthermore, as an application we establish k -fractional calculus operators for the novel matrix function. We also give some new and known results as special cases of our proposed generalization of the Wright hypergeometric matrix function.

On New Matrix Version Extension of the Incomplete Wright Hypergeometric Functions and Their Fractional Calculus

Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric matrix function and state some integral equations and differential formulas about it. Next, we obtain some results about fractional calculus of these extended incomplete Wright hypergeometric matrix functions. Finally, we discuss an application of the extended incomplete Wright hypergeometric matrix function in the kinetic equations.

On certain Kampé de Fériet-like hypergeometric matrix functions

In this paper, we introduce certain Kampé de Fériet-like matrix functions and investigate their regions of convergence and recurrence relations satisfied by them. The importance of studying these matrix functions stems from the fact that the derivatives of arbitrary integral order of confluent hypergeometric matrix function and Gauss hypergeometric matrix function with respect to their matrix parameters can be expressed in the form of these Kampé de Fériet-like matrix functions. Integral representations of these matrix functions in a special case are also given.

A study of generalized hypergeometric Matrix functions via two-parameter Mittag–Leffler matrix function

Abstract The main aim of this article is to study a new generalizations of the Gauss hypergeometric matrix and confluent hypergeometric matrix functions by using two-parameter Mittag–Leffler matrix function. In particular, we investigate certain important properties of these extended matrix functions such as integral representations, differentiation formulas, beta matrix transform, and Laplace transform. Furthermore, we introduce an extension of the Jacobi matrix orthogonal polynomial by using our generalized Gauss hypergeometric matrix function, which is very important in scattering theory and inverse scattering theory.

Contribution of Using Hadamard Fractional Integral Operator via Mellin Integral Transform for Solving Certain Fractional Kinetic Matrix Equations

Recently, the importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in electrodynamics, control systems, economic, physics, geophysics and hydrodynamics. Among the many fractional differential equations are kinetic equations. Fractional-order kinetic Equations (FOKEs) are a unifying tool for the description of load vector behavior in disorderly media. In this article, we employ the Hadamard fractional integral operator via Mellin integral transform to establish the generalization of some fractional-order kinetic equations including extended (k,τ)-Gauss hypergeometric matrix functions. Solutions to certain fractional-order kinetic matrix Equations (FOKMEs) involving extended (k,τ)-Gauss hypergeometric matrix functions are also introduced. Moreover, several special cases of our main results are archived.

On degenerate gamma matrix functions and related functions

Recently, the applications of special functions of matrix arguments have received more attention in many fields, such as theoretical physics, number theory, probability theory, engineering and theory of group representations. Gaining enlightenment from these works, in this paper, we introduce the degenerate gamma matrix function, the degenerate zeta matrix function, the degenerate digamma matrix function, the degenerate polygamma matrix function and the degenerate Gauss hypergeometric matrix function. Basic properties of these functions are discussed. In addition, we derive some interesting formulas related to these functions and special cases.

Further investigations on the two variables second Appell hypergeometric matrix function

Over the last two decades, special matrix functions have become a major area of study for mathematicians and physicists. The famous four Appell hypergeometric matrix functions have received considerable attention by many authors from different points of view. The present paper is devoted to provide further investigations on the two variables second Appell hypergeometric matrix functions. Precisely, we derive certain mathematical properties such as limit formulae of Humbert hypergeometric matrix functions, recursion formulae using contiguous matrix relations, some differentiation and summation formulae. We also find general differential operators which induce some differential equations. This enriches the theory of special matrix functions. The obtained results are believed to be newly presented.