Saigo Operators Research Articles

Marichev-Saigo-Maeda Fractional Integration of Product of the K- Function and the -Function

Fractional calculus is an important branch of mathematical analysis which deals with investigations of integrals and derivatives of arbitrary order. It has been applied to almost every field of Science, Engineering and Mathematics. The aim of this paper is to study the generalized Marichev- Saigo- Maeda fractional integral operators. We will establish two theorems which give the image of the product of H-function and K-function in Saigo operators. Certain Special cases of main results are also discussed.

Nonlocal Problem With Saigo Operators for Mixed Type Equation of the Third Order

For a third order equation with multiple characteristics we investigate a boundary-value problem with Saigo operators. We prove the unique solvability of the problem for various values of the parameters of generalized fractional integro-differentiation operators

Нелокальная задача с операторами Сайго для уравнения смешанного типа третьего порядка

Нелокальная задача с операторами Сайго для уравнения смешанного типа третьего порядка

Boundary-Value Problem With Saigo Operators for Mixed Type Equation With Fractional Derivative

We set up and solve a non-local problem for a differential equation, which contains the diffusion equation of fractional order. The boundary condition contains a linear combination of generalized operators with the Gauss hypergeometric function in the kernel. For various values of parameters of these operators we write a solution in explicit form.

On Generalized Fractional Integration of Aleph $$(\aleph )$$ ( ℵ ) -Function

In the present paper, we establish two theorems exhibiting images of the product of Aleph \((\aleph )\)-function and Srivastava’s polynomials (J Math 14:1–6, 1972) in Saigo operators (Math Rep Kyushu Univ 11:135–143, 1978). Being of general and unified in nature, the results established here yield a large number of new and known images involving Riemann–Liouville and Erd’elyi–Kober fractional integral operators and several special functions as special cases. To illustrate, four corollaries have been recorded here.

Erdélyi-Kober fractional integral operators from a statistical perspective (I)

In this article we examine the densities of a product and a ratio of two real positive scalar random variables $x_1$ and $x_2$, which are statistically independently distributed, and we consider the density of the product $u_1=x_1x_2$ as well as the density of the ratio $u_2=\frac{x_2}{x_1}$ and show that Kober operator of the second kind is available as the density of $u_1$ and Kober operator of the first kind is available as the density of $u_2$ when $x_1$ has a type-1 beta density and $x_2$ has an arbitrary density. We also give interpretations of Kober operators of the second and first kind as Mellin convolution for a product and ratio respectively. Then we look at various types of generalizations of the idea thereby obtaining a large collection of operators which can all be called generalized Kober operators. One of the generalizations considered is the pathway idea where one can move from one family of operators to another family and yet another family and eventually end up with an exponential form. Common generalizations in terms of a Gauss' hypergeometric series is also given a statistical interpretation and put on a more general structure so that the standard generalizations given by various authors, including Saigo operators, are given statistical interpretations and are derivable as special cases of the general structure considered in this article.

ОБ ОДНОЙ ЗАДАЧЕ С ОБОБЩЁННЫМИ ОПЕРАТОРАМИ ДРОБНОГО ДИФФЕРЕНЦИРОВАНИЯ ДЛЯ ВЫРОЖДАЮЩЕГОСЯ ВНУТРИ ОБЛАСТИ ГИПЕРБОЛИЧЕСКОГО УРАВНЕНИЯ

In this paper, we consider a problem with Saigo operators of fractional differentiation in a boundary condition on a characteristic part of a boundary. The unique solvability of this problem is proved.

FRACTIONAL DIFFERENTIATION OF THE PRODUCT OF APPELL FUNCTION F3AND MULTIVARIABLE H-FUNCTIONS

Fractional calculus operators have been investigated by many authors during the last four decades due to their importance and useful- ness in many branches of science, engineering, technology, earth sciences and so on. Saigo et al. (9) evaluated the fractional integrals of the product of Appell function of the third kernel F3 and multivariable H-function. In this sequel, we aim at deriving the generalized fractional differentia- tion of the product of Appell function F3 and multivariable H-function. Since the results derived here are of general character, several known and (presumably) new results for the various operators of fractional dif- ferentiation, for example, Riemann-Liouville, Erdelyi-Kober and Saigo operators, associated with multivariable H-function and Appell function F3 are shown to be deduced as special cases of our findings.

Boundary-value problem with saigo operators for mixed type equation of the third order with multiple characteristics

We study a boundary-value problem with shift formixed-type equation of the third order. In the hyperbolic field boundary condition contains a linear combination of generalized operators of fractional integro-differentiation. We prove a unique solvability of the problem.

A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type

We consider a mixed-type equation whose order degenerates along the line of change of type. For this equation we study the unique solvability of a nonlocal problem with the Saigo operators in the boundary condition. We prove the uniqueness theorem under certain conditions (stated as inequalities) on known functions. To prove the existence of solution to the problem, we equivalently reduce it to a singular integral equation with the Cauchy kernel. We establish a condition ensuring the existence of a regularizer which reduces the obtained equation to a Fredholm equation of the second kind, whose unique solvability follows from that of the problem.

ON A PROBLEM WITH GENERALIZED OPERATORS OF FRACTIONAL DIFFERENTIATION FOR A DEGENERATED INSIDE A DOMAIN HYPERBOLIC EQUATION

In this paper, we consider a problem with Saigo operators of fractional differentiation in a boundary condition on a characteristic part of a boundary. The unique solvability of this problem is proved.

Fractional Integration of the Product of Two H-functions and a General Class of Polynomials

The aim of the present paper to study and develop the generalized fractional integral operators recently introduce by Saigo [5]. First, author establish two results that give the image of the products of two H-functions and a general class of polynomials in Saigo operators. These results, besides being of very general character have been put in a compact form avoiding the occurrence of in…nite series and thus making them useful in applications. Our …ndings provide interesting uni…cations and extensions of a number of (new and known) images. For the sake of illustration, we give here exact references to the results (in essence) of eight research papers [1, 2, 3, 4, 7, 8, 10 and 11] that follow as particular cases of our …ndings.

The two dimensional generalized Weyl fractional calculus and special functions

The object of this present paper is to derive a relation between the two dimensional I-transform involving a general class of polynomials and the Weyl type two dimensional Saigo operators of fractional integration. The results derived here are general in nature and include the results given earlier by Saigo, Saxena and Ram [10],Saxena and Ram [8], Saxena and Kiryakova [9] and Chaurasia and Srivastava [12].

Fractional calculus of the $ bar H $-function

The subject of this paper is to derive a fractional calculus formula for $ bar H $-function due to Inayat-Hussain whose based upon generalized fractional integration and differentiation operators of arbitrary complex order involving Appell function $F_3$ due to Saigo & Meada. The results are obtained in a compact form containing the Reimann-Liouville, Eredlyi-Kober and Saigo operators of fractional calculus.

Distortion and Convolutional Theorems for Operators of Generalized Fractional Calculus Involving Wright Function

Using the Wright's generalized hypergeometric function, we investigate a class W(q, s; A, B, λ) of analytic functions with negative coefficients. We derive many results for the modified Hadamard product of functions belonging to the class W(q, s; A, B, λ). Moreover, we generalize some of the distortion theorems to the classical fractional integrals and derivatives and the Saigo (hypergeometric) operators of fractional calculus.