Maged G. Bin-Saad

Aden University · Mathematics

In this paper, we stress the importance of the Mittag–Leffler function of two parameters and a single variable in the framework of mathematical physics and applied mathematics. We begin with pseudo hyperbolic and trigonometric functions and progress to introduce an arbitrary order Mittag–Leffler-type function. We study its properties, basic relatio...

Our fffindings and discussions will add to the body of knowledge on Mittag-Leffer functions. A new extended bivariate Mittag-Leffer function is being studied in this work, with numerous well-known Mittag-Leffer functions acting as special instances. Its characteristics, integral and operational relationships, fractional calculus, pure and differen-...

Our findings and discussions will add to the body of knowledge on Mittag-Leffler functions. A new extended bivariate Mittag-Leffler function is being studied in this work, with numerous well-known Mittag-Leffler functions acting as special instances. Its characteristics, integral and operational relationships, fractional calculus, pure and differen...

We study bivariate Mittag-Leffler function which is an extension of several known Mittag-Leffler functions. We derive the Riemann-Liouville fractional derivatives and integrals of these function and solve a singular integral equation with the bivariate Mittag-Leffler function in the kernel. We also introduce and investigate a fractional integral op...

In this paper, we exploit the operational identities of the Burchnall-type and monomiality principle to introduce and discuss a new class of 2D-complex Hermite polynomials. We study their properties (raising and lowering operational relations, Burchnall's operational formulas, differential equations, generating functions, recurrence relations, expa...

Based on the generalized extended beta function, we shall introduce and study a new hypergeometric-type extended zeta function together with related integral representations , differential relations, finite sums, and series expansions. Also, we present a relationship between the extended zeta function and the Laguerre polynomials. Our hypergeometri...

By extended beta function, this paper investigates an extended form HurwitzLerch zeta function together with related integral images, ordinary and fractional derivatives, and series expansions. Also, we introduce a relationship between the extended HurwitzLerch zeta function and the Laguerre polynomials. Further, we present an application of the ex...

The present paper establishes several new integral representations of the Euler type and Laplace type for some Gauss hypergeometric functions of three variables. The main results are obtained by using the properties of Gamma and beta functions. The novel integral representations are carried out through ten hypergeometric functions of three variable...

The principal object of this paper is to present new contractive type condition for mappings defined on G-metric spaces. Further, we prove some new fixed point theorems concerning these mappings in G-metric space. Particular cases and examples to illustrate and support our results are also considered. Our findings extend and unify a known results....

The purpose of this paper is to get new generating relations of the complex Hermite polynomials $H_{p,q}\left ( z,z^{*} \right )$ by extending the realization $\uparrow_{\omega,\mu}$ to study multiplier representations of a Lie group G(0,1). Also, we establish new operational formulas involving the polynomials $H_{p,q}\left ( z,z^{*} \right )$ and...

Generating functions plays an essential role in the investigation of several useful properties of the sequences which they generate. In this paper, we establish certain generating relations, involving some quadruple hypergeometric functions introduced by Bin-Saad and Younis. Some interesting special cases of our main results are also considered.

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integ...

The hypergeometric series of four variables are introduced and studied by Bin-Saad and Younis recently. In this line, we derive several fractional derivative formulas, integral representations and oper- ational formulas for new quadruple hypergeometric series.

This paper deals with the fundamental solutions for a two-dimensional elliptic equations with two singular coefficients. we construct the fundamental solutions of generalized axially symmetric Helmholtz equation in terms of a confluent hypergeometric function in two variables.

It is shown that an appropriate combination of methods, relevant to operational calculus and to special functions, can be a very useful tool to establish and treat a new modified 2D-Laguerre polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new modified 2D-Laguerre polynomials and di...

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integ...

The main aim of this present paper is to present certain generating functions of some hypergeometric functions in four variables by using the integral and symbolic representations for these quadruple functions. A few interesting special cases have also been considered.

We introduce new quadruple hypergeometric functions, which is denoted by X^ (4)_{i}(i = 61,62,...,69); and investigate several interesting integral representations of Euler-type and Laplace-type for quadruple hypergeometric functions.

Recently, Casadei [4] provided an explicit formula for statistical marginal model in terms of Poisson-Gamma mixture. This model involving certain polynomials which play the key role in reference analysis of the signal and background model in counting experiments. The principal object of this paper is to present a natural further step toward the mat...

In the present work, we establish a new integral representations of Euler type for the quadruple hypergeometric functions X ^(4)_ i (.), i = 38, 40, 45, 48, 50. We obtain some operational relations include these quadruple functions .

It is shown that an appropriate combination of methods, relevant to operational calculus and to special functions, can be a very useful tool to establish and treat a new modified 2D-Laguerre polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new modified 2D-Laguerre polynomials and di...

The main objective of this work is to introduce a new generalization of Hurwitz-Lerch zeta function of two variables. Also, we investigate several interesting properties such as integral representations, operational connections and summation formulas.

Hypergeometric function of four variables was introduced by Bin-Saad and Younis. In the present paper a new integral representations of of Euler-type and Laplace-type involving double and triple hypergeometric series for these functions are derived.

In this paper, we introduce ten new quadruple hypergeometric series. We also obtain their various properties such that integral representations, fractional derivatives, N-fractional connections, operational relations and generating functions.

A new class of quadruple hyper-geometric series is presented. We also give integral representations of Euler-type and Laplace-type for the new class of series.

The main objective of this work is to introduce a new generalization of Hurwitz-Lerch zeta function of two variables. Also, we investigate several interesting properties such as integral representations, operational connections and summation formulas.

While investigating the Exton's list of twenty one hyper-geometric functions of four variables and the Sharma's and Parihar's list of eighty three hyper-geometric functions of four variables, we noticed existence of new hyper-geometric series of four variables. The principal object of this paper is to introduce new hyper-geometric series of four va...

In this article, we first give some basic properties of generalized Her- mite polynomials associated with parabolic cylinder functions. We next use Weisnerś group theoretic method and operational rules method to establish new generating functions for these generalized Hermite polynomials. The operational methods we use allow us to obtainunilateral,...

We aim in this work at establishing interesting operational connections between new quadruple hypergeo-metric series) 30 , , 1 () 4 ( = i X i defined in [9] and certain class of triple series involving of Exton's functions 1 X to 20

we aim in this work at establishing interesting operational connections between new quadruple hypergeometric series defined in [9] and certain class of triple series involving of Exton’s functions to , Srivastava's functions , Lauricella's functions and the general triple hypergeometric series . Some particular cases and consequences of our main re...

In this article, we first give some basic properties of generalized Hermite poly-nomials associated with parabolic cylinder functions. We, next use Weisner´sWeisner´s group theoretic method and operational rules method to establish new generating functions for these generalized Hermite polynomials. The operational methods we use allows us to obtain...

In this work we aim at establishing certain generating functions, involving the quadruple functions X (4) 9 and X (4) 10 defined in [1], where everyone can be represented by Laplace integral. Some particular cases and consequences of our main results are also considered.

The main objective of this paper is to present integral representations of Euler type and Laplace type for five new hypergeometric series of four variables.

While investigating the Exton's list of twenty one hyper-geometric functions of four variables and the Sharma's and Parihar's list of eighty three hyper-geometric functions of four variables, we noticed existence of new hyper-geometric series of four variables. The principal object of this paper is to introduce new hyper-geometric series of four va...

In this paper, we define some new quadruple hypergeometric functions , which we denoted by X^ (4)_ i (i = 31, 32, ..., 50). Then, we obtain its integral representations of Euler-type and Laplace-type.

In this work, it is shown that the combination of operational techniques and the use of the principle of quasi-monomiality can be a very useful tool for a more general insight into the theory of matrix polynomials and for their extension. We explore the formal properties of the operational rules to derive a number of properties of certain class of...

In this work, by using Laplace integral representation of quadruple function X^(4)_8 defined in [1], we introduce new generating functions involving some triple hyper-geometric functions and X^(4)_8 itself. Some particular cases and consequences of our main results are also considered. Cite This Article: Maged G. Bin-Saad

In this work, it is shown that the combination of operational techniques and the use of the principle of quasi-monomiality can be a very useful tool for a more general insight into the theory of matrix polynomials and for their extension. We explore the formal properties of the operational rules to derive a number of properties of certain class of...

In this work we aim at establishing certain generating functions, involving the quadruple functions X (4) 9 and X (4) 10 defined in [1], where everyone can be represented by Laplace integral. Some particular cases and consequences of our main results are also considered.

We aim in this work to establish new operational representations for the hypergeometric functions of four variables X^(4)_6, X^(4)_7, X^(4)_8, X^(4)_9, X^(4)_10. By means of these operational representations, a number of generation functions involving these hypergeometric functions are then found.

In this work, by using Laplace integral representation of quadruple function X^(4)_ 7 defined in [3], we introduced new generating functions involving some quadruple hypergeometric functions. Some particular cases and consequences of our main results are also considered.

The purpose of this work is to give modified definitions of Apostol-Euler polynomials and numbers of higher order. We establish their elementary properties, sums, explicit relations, integrals and differential relations. A number of new results which introduced are generalization of known results and their special cases lead to the corresponding fo...

Based upon the classical derivative and integral operators we introduce a new symbolic operational representations for the hypergeometric function of four variables (4)F_30. By means of these symbolic operational representations number of generating functions involving the hypergeometric function (4)F_30 are then found. Some special cases of the ma...

In this paper, we define some new quadruple hypergeometric functions, which we denoted by X^ (4)_ i (i = 31, 32, ..., 50). Then, we obtain its integral representations of Euler-type and Laplace-type.

The purpose of this work is to give modified definitions of Apostol-Euler polyno-mials and numbers of higher order. We establish their elementary properties, sums, explicit relations, integrals and differential relations. A number of new results which introduced are generalization of known results and their special cases lead to the corresponding f...

The purpose of this paper is to introduce and investigate a new class of multiple zeta functions of variables. We study its properties , integral representations, differential relation, series expansion and discuss the link with known results.

Fractional calculus has emerged as one of the most important interdisciplinary subjects during the last four decades mainly due to its applications in various fields of science and engineering. In this paper we exploit the fractional calculus to discuss a new class of Laguerre polynomials of two fractional ( arbitrary ) orders. The properties of th...

It is shown that an appropriate combination of methods, relevant to matrix polynomials and to operational calculus can be a very useful tool to establish and treat a new class of matrix Laguerre–Konhauser polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new class of Laguerre–Konhaus...

It is shown that an appropriate combination of methods, relevant to matrix polynomials and to operational calculus can be a very useful tool to establish and treat a new class of matrix Laguerre-Konhauser polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new class of Laguerre-Konhaus...

This paper is sequel to the authors paper [18]. By using the generalized Burchnall-Chaundy operator method, the authors are aiming at deriving certain decomposition formulas for some interesting special cases of Kampé de Fériets series of double hypergeometric series F p:q;k l:m;n .

We aim in this work at establishing new integral representations of Euler type for the Exton hypergeometric functions of four variables K6,K7,K8,K9,K10, the Sharma and Parihar hypergeometric functions of four variables (4)F7,(4)F41,(4)F61and the Lauricella function of four variables (4)FA, whose kernels include the quadruple hypergeometric function...

we aim in this work at establishing interesting operational connections between new quadruple hyper-geometric series) 30 , , 1 () 4 (  i X i defined in [9] and certain class of triple series involving of Exton's functions 1

The main object of this work is to introduce a new multivariable extension of the Hurwitz-Lerch Zeta function. We then systematically investigate its mathematical properties and give its explicit relationship with new defined Apostol- Euler polynomials of several variables. We also, consider some important special cases.

In the present paper, we establish some interesting integrals involving the product of Bessel matrix functions of the first kind with Jacobi matrix polynomials, which are expressed in terms of the matrix version Kampe' de Fe'riet and Srivastava and Daoust functions. Some other integrals involving the product of the matrix version of Bessel matrix f...

In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equa- tions satisfied by hyper-geometric functions and find explicit lin- early independent solutions for the system. In this investigation, we give the solution...

Horn [1931, Hypergeometrische Funktionen zweier Veranderlichen, Math. Ann., 105(1), 381-407], (corrections in Borngasser [1933, Uber hypergeometrische funkionen zweier Ve- randerlichen, Dissertation, Darmstadt], defined and investigated ten hypergeometric series in two variables and of order two. In the course of further investigation of Horn’s ser...

In our present investigation we propose to present certain decomposition formulas for Kamp? de F?riets series of double hypergeometric series F p : q ; k l : m ; n . Based upon the theory of symbolic operators, we introduce a new symbolic operational images and by means of these symbolic operational images a number of decomposition formulas for Kam...

While investigating the Exton's list of 21 hypergeometric functions of four variables namely K 1 , K 2 ,. .. , K 21 and the Sharma's and Parihar's list of eighty three hypergeometric functions of four variables namely F (4) 1 , F (4) 2 ,. .. , F (4) 83 , we noticed the existence of new hypergeometric series of four variables. In the present work, w...

In this work, it is shown that the combination of operational techniques and the use of the principle of quasi-monomiality can be a very useful tool for a more general insight into the theory of matrix polynomials and for their extension. We explore the formal properties of the operational rules to derive a number of properties of certain class of...

The main purpose of this paper is to introduce the matrix extension of the pseudo- Laguerre matrix polynomials and to explore the formal properties of the opera- onal rules and the principle of quasi-mono

The principal object of this paper is to present a natural further step toward the mathematical properties and presentations concerning the two variable Laguerre matrix polynomials defined in (Bin-Saad, Maged G., Antar, A. Al-Sayaad, 2015. Study of two variable Laguerre polynomials via symbolic operational images. Asian J. of math. and comput. rese...

The present paper aims at harnessing the technique of Lie Algebra and operational methods to derive and interpret generating relations for the three- variable Hermite Polynomials H n (x,y,z) introduced recently in [2]. Certain gener- ating relations for the polynomials related to H n (x,y,z) are also obtained as special cases.

The principal object of this paper is to present a natural further step toward the mathematical properties and presentations concerning the two variable Laguerre matrix polynomials defined in (Bin-Saad, Maged G., Antar, A. Al-Sayaad, 2015. Study of two variable Laguerre polynomials via symbolic operational images. Asian J. of math. and comput. rese...

New generalized form of the multi-variable Gegenbauer matrix polynomials are introduced using the integral representation method. Certain properties for these new generalized multi-variable Gegenbauer matrix polynomials such as differential relations, operational and hypergeometric matrix representations, generating matrix functions are derived. Fu...

In this work we will introduce and study a generalized zeta function of the general Hurwitz – Lerch zeta functions. Here, we aim to derive basic properties of generalized zeta function for the general Hurwitz – Lerch zeta functions include some integral representations for several general Hurwitz – Lerch zeta functions and fractional derivative. A...

In this work, we obtain certain integral representations for functions related to Kampe' de Fe'riet function of the fourth order ,which are the sufficiently general in nature and are capable of yielding a large number of simpler and useful results merely by specializing the parameters in them.

This work sequel to our paper [2] and we aim to obtain certain integral representations for functions related to Kampe' de Fe'riet function of the fourth order ,which are the sufficiently general in nature and are capable of yielding a large number of useful and simpler results merely by specializing the parameters between them.

The subject of fractional calculus has gained importance and popularity during the past three decades. Based upon the N-fractional calculus we introduce a new N-fractional operators involving hyper-geometric function. By means of these N- fractional operators a number of operational relations among the hyper-geometric functions of two, three, four...

We obtain here certain integral representations for functions related to Kampe' de Fe'riet function of the fourth order ,which are the sufficiently general in nature and are capable of yielding a large number of simpler and useful results merely by specializing the parameters in them.

Abstract. In this paper, an extension of the Hermite matrix polynomials is introduced. Some relevant matrix functions appear in terms of the two-index and two-variable and p-index and p-variable Hermite matrix polynomials. Furthermore, in order to give qualitative properties of this family of matrix polynomials, the Legendre and Chebyshev matrix po...

The present work aims at introducing and investigating a new two classes of hypergeometric-type generating functions associated with general family of Hurwitz-Lerch zeta functions and derives their basic properties including integral representations, sums and fractional calculus and discuss the link with known results.

Based upon the classical derivative and integral operators we introduce a new symbolic operational images for hypergeometric functions of two and three variables. By means of these symbolic operational images a number of operational relations among the hypergeometric functions of two and three variables are then found. Other closely-related results...

This paper refers to some generalizations of certain classical Rodrigues formulas. By means of the Riemann - Liouville operator of fractional calculus general Rodrigues-type representation for- mulas of fractional order are derived and some of their properties are given and compared with the corresponding properties of known cases.

Based upon the classical derivative and integral operators we introduce a new symbolic operational images for hypergeometric functions of two and three variables. By means of these symbolic operational images a number of operational relations among the hypergeometric functions of two and three variables are then found. Other closely-related resu...

It is shown that an appropriate combination of methods, relevant to operational calculus and to matrix polynomials, can be a very useful tool to establish and treat a new class of two variable Laguerre matrix polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new class two variable La...

The principal object of this paper is to study a class of matrix poly- nomials associated with Humbert polynomials. These polynomials generalize the well known class of Gegenbauer, Legendre, Pincherl, Horadam, Horadam-Pethe and Kinney polynomials. We shall give some basic relations involving the Humbert matrix polynomials and then take up several g...

The authors derive a general theorem on multidimensional generating func-tions involving arbitrary coefficients. By appropriately specializing these coefficients a number of (known and new) results are shown to follow as applications of the theorem.

In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. Here we choose the Exton function $K_{2}$ among...

The aim of this work is to derive general integral formulas involving Laguerre - Konhauser and Bessel polynomials which appear to be new. The approach is beginning with an expression for the product of two Laguerre-Konhauser polynomials and two Bessel functions to integrate and perform a re-summation to obtain other functions, and then to reduce th...

An attempt is made here to introduce and study a class of hypergeometric-type generating function associated with multiple zeta functions together with related integral representations, differential relations and sums. A number of (known and new) results shown to follow as special cases of our theorems.

The authors derive a general theorem on multidimensional generating func-tions involving arbitrary coefficients. By appropriately specializing these coefficients a number of (known and new) results are shown to follow as applications of the theorem.

Based upon the classical derivative and integral operators we introduce a new operator which allows the derivation of new symbolic operational images for hypergeometric functions. By means of these symbolic operational images a number of decomposition formulas involving quadruple series are then found. Other closely-related results are also conside...

It is shown that an appropriate combination of methods, relevant to opera- tional calculus and to special functions, can be a very useful tool to establish and treat a new class of Hermite and Konhauser polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new class of Hermite and Konhau...

The principal object of this paper is to present a natural further step toward the mathematical properties and presentations concerning the generalized Voigt functions. Recurrence relations, connections, series expansions and integral representations involving classical functions of mathematical physics and hypergeometric series for these functions...

The present work is a sequel to the papers [3] and [4], and it aims at introducing and investigating a new generalized double zeta function involving the Riemann, Hurwitz, Hurwitz-Lerch and Barnes double zeta functions as particular cases. We study its properties, integral representations, differential relations, series expansion and discuss the li...

Using the method of N-fractional calculus, representations of certain triple hypergeometric functions are established. These fractional representations play a key role in obtaining transformation and reduction formulae of these triple hypergeometric functions. Some known results of Nishimoto, Erdelyi and Exton are deduced as special cases of our fo...

An attempt is made here to introduce and study a pair of double power series associated with the generalized zeta function due to Erdélyi Φ(x, z, a) together with related sums, integral representations, generating relations and N -fractional calculus. A number of (known and new) results shown to follow as special cases of our theorems.

The authors prove a general theorem on multidimensional generating relations. As applications of this general theorem, a number of (known or new) genreating functions of several variables are deduced. Connections of the various generating functions, which are presented in this paper, with several known results are also considered.

In this paper we exploit the monomiality principle to discuss and introduce a new class of Laguerre–Konhauser polynomials. We study their properties (differential equations, generating functions, recurrence relations, expansions and so on), and discuss the link with ordinary case.

In this paper, we aim at introducing and studying two hypergeometric-type generating functions associated with the generalized zeta function; our goal is to derive their basic properties including integral representations, sums, series representations and generating functions. A number of (known and new) results are shown to follow as special cases...

There are several known methods for obtaining transformations and cases of reducibility of hypergeomatric series in two and more variables. In this paper, we aim at introducing a new application of the N-fractional calculus as a method for obtaining transformations formulae for multi variable hypergeomatric series. We also show how a number of (new...

There are several known methods for obtaining transformations and cases of reducibility of hypergeomatric series in two and more variables. In this paper, we aim at introducing a new application of the N-fractional calculus as a method for obtaining transformations formulae for multi variable hypergeomatric series. We also show how a number of (new...

It is often convenient to identify the various functions with contour integral along certain paths in the complex plane. These integrals provide recursion formulas, asymptotic forms and analytic continuations of the special functions. In this paper we consider N-fractional representations for Appell’s function F1 and obtained a number of special ca...

The authors derive a general theorem on partly bilateral and partly unilateral generating functions involving multiple series with essentially arbitrary coefficients. By appropriately specialising these coefficients, a number of (known or new) results are shown to follow as applications of the theorem.

The present paper aims at deriving new relations on bilinear and bilateral generating functions involving double and triple series with essentially arbitrary coefficients. We show a number of ( known and new) results can be deduced from main generating relations by appropriately specializing the coefficients involved.

The aim of this paper is to derive a theorem on partly bilateral and partly unilateral generating functions. A number of )known and new ) results follows as special cases, simply by specializing the cofficients and parameters involved in the theorem. For the sake of illustration, some special cases are mentioned briefly.