arXiv subjects
Publications and source records attributed to Ayman Shehata.
The aim of the present study is to establish some properties for q-Bessel matrix polynomials such as several q-differential matrix equation, q-differential matrix relations and q-recurrence matrix relations, and integral representation, q-Laplace and q-Mellin transforms with the help of q-Analysis. Furthermore, we give connections between q-Horn's matrix functions of two variables and q-Bessel matrix polynomials are given.
The study is devoted to the construction of dynamical symmetry algebra of confluent hypergeometric function $_1F_1$ and $\Psi_2$-Humbert function and to derive some generating relations and reduction formulas for $_1F_1$ and $\Psi_2$ functions.
The object of this paper is to investigate the certain results involving Bateman's matrix polynomials for integral index. We obtain some properties, integral representation and recurrence relations for hypergeometric matrix function. We introduce some matrix differential equations of the three order, integral transform and fractional integral formulas for hypergeometric matrix function by using the beta and Laplace transforms formula, Erdelyi Kober type fractional integral operators
Recently, extensions of gamma and beta functions have been studied by many researchers due to their nice properties and variety of applications in different fields of science. The aim of this note is to investigate generalized inequalities associated with extended beta and gamma functions
Recently, Shehata et al. [37] introduced the $_{r+1}R_{s,k}(B,C,z)$ matrix function and established some properties. The aim of this study established to devote and derive certain basic properties including analytic properties, recurrence matrix relations, differential properties, new integral representations, $k$-Beta transform, Laplace transform, fractional k-Fourier transform, fractional integral properties, the $k$-Riemann-Liouville and $k$-Weyl fractional integral and derivative operators an extended version of $_{r+1}R_{s,k}$ matrix function. We establish its relationships with other well known special matrix functions which have some particular cases in the context of three parametric Mittag-Leffer matrix function, $k$-Konhauser and $k$-Laguerre matrix polynomials. Finally, some special cases of the established formulas are also discussed.
This note introduces a new range of modified gamma and beta $k$ functions. The authors present new modified gamma and beta $k$-functions, first and second summation relations, various functionals, Mellin transforms, and integral representations. Furthermore, mean, variance and the moment generating function of a generalized beta distribution are obtained.
The recurrence matrix relations, differentiation formulas, and analytical and fractional integral properties of incomplete gamma matrix functions $γ(Q, x)$ and $Γ(Q, x)$ are all covered in this article. The generalized incomplete exponential matrix functions with their integral representations functions have been examined, along with some relevant characteristics of these functions such as integral representations functions . Additionally, the infinite summation relations and formulas for two sequences are shown, along with the generalized incomplete exponential matrix functions with the integral representation, addition formula for addition of two arguments, multiplication formula for multiplication of two arguments, and recurrence matrix relation.
The main aim of this work is to derive the $q$-recurrence relations, $q$-partial derivative relations and summation formula of bibasic Humbert hypergeometric function $Φ_1$ on two independent bases $q$ and $q_{1}$ of two variables and some developments formulae, believed to be new, by using the conception of $q$-calculus.
The main aim of the present work is to give some interesting the $q$-analogues of various $q$-recurrence relations, $q$-recursion formulas, $q$-partial derivative relations, $q$-integral representations, transformation and summation formulas for bibasic Humbert hypergeometric functions $Ψ_{1}$ and $Ψ_{2}$ on two independent bases $q$ and $p$ of two variables and some developments formulae, believed to be new, by using the conception of $q$-calculus. Finally, some interesting special cases and straightforward identities connected with bibasic Humbert hypergeometric series of the types $Ψ_{1}$ and $Ψ_{2}$ are established when the two independent bases $q$ and $p$ are equal.
This paper studies derivatives with respect to the parameters of Srivastava triple hypergeometric functions HA, HB and HC. Using basic properties of the Gamma function and Pochhammer symbols, we obtain explicit formulas for first and higher order derivatives. These derivatives are expressed in terms of Pathan quadruple hypergeometric function F. We also derive Euler type differential operator identities, contiguous relations for unit shifts in the parameters, and recurrence relations satisfied by these derivatives. In addition, we show that derivatives of arbitrary order satisfy systems of linear partial differential equations in the underlying variables. The results extend known differentiation formulas for classical and multivariable hypergeometric functions and provide tools for potential applications in mathematical physics and engineering.
In this article, we investigate and establish some properties including analytic properties, contiguous relations, differential properties, differential operators, an expansion formula, and simple integrals, integral operators, some fractional integral properties, some new integral representations, the Riemann Liouville fractional $q$-derivative and $q$-integral operators of $q$-analogue of various basic $_{r}R_{s,q}$ function by using technique of $q$-calculus. Certain interesting consequences of the theorem are also discussed by considering some examples.
Given the growing quantity of proposals and works of basic hypergeometric functions in the scope of $q$-calculus, it is important to introduce a systematic classification of $q$-calculus. Our aim in this article is to investigate certain interesting several $q$-partial derivative formulas, $q$-contiguous function relations, $q$-recurrence relations, various $q$-partial differential equations, summation formulas, transformation formulas and $q$-integrals representations for basic Humbert confluent hypergeometric functions under what constraints of parameters. These interesting properties, as special cases, include many known expansions of basic Humbert hypergeometric functions, and are also include particular interest in the area.
Humbert confluent hypergeometric functions of two variables arise in many problems of mathematical physics and applied analysis, yet their behavior with respect to parameters has not been systematically studied. In this paper we investigate derivatives with respect to numerator and denominator parameters for the seven classical Humbert functions \Phi{1}, \Phi{2}, \Phi_{3}, Psi_{1}, Psi_{2}, \Xi_{1} and \Xi_{2}. Using their double series representations together with elementary properties of the Gamma and digamma functions, we derive explicit formulas for first order parameter derivatives and express them in compact form in terms of Srivastava triple hypergeometric function F{3}. By differentiating the underlying partial differential equations, we further obtain simple operator recurrences for derivatives of arbitrary order, which yield closed differentiation and reduction formulas in terms of contiguous Humbert functions. Finally, we indicate how these results lead to Taylor type parameter expansions and illustrate their use with basic numerical examples and plots.