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On Three-Dimensional Saigo-Maeda Operator of Weyl Type with Three Dimensional H-Transforms

Singh D^1* and Jain, R^{2 *}Correspondence: Singh D, Department of Applied Mathematics, School of Sciences, ITM University, India, Email:

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Introduction

Saxena, Gupta and Kumbhat [16] studied two dimensional Weyl fractional calculus. Nishimoto and Saxena [8] gave the generalization of results given by Arora, Raina and Koul [1], Saxena, Gupta and Kumbhat [16], and Raina and Kiryakova [12] by proving the theorems associated with two dimensional G-transforms. Saxena, Ram and Goree [14] proved a theorem on two dimensional generalized H-transforms involving Weyl type two dimensional Saigo operators. Saxena, Ram and Suthar [13] studied the generalized fractional integration operators associated with the Appell Function F₃ as kernel, introduced recently by Saigo and Maeda [10]. Chaurasia and Srivastava [19] established a theorem on two dimensional -transforms involving polynomials of general class with Weyl type two dimensional Saigo operators. Chaurasia and Jain [18] evaluated certain triple integral relations, and derived a theorem on three dimensional -transforms associated with the three dimensional Saigo operators of Weyl type.

In this paper we drive a relationship between H-transforms and the Saigo-Maeda operators of Weyl type of three dimensions. The results obtained here provide extension of Saigo results, Saxena and Ram [9], Saxena and Ram [12] and Chaurasia and Jain [18].

DEFINITIONS: The Fox’s H-function [2, 3, 4] is in the form of Mellin-Barnes type [7] integral in following way

Where 0 ≤ n ≤ p, 0 ≤ m ≤ q are non-negative integers; A_j and B_j are positive; a_j ( j=1,…,p) and b_j (j=1,…, q) are real or complex, such that A_j ( b_h + ν ) ≠ B_h ( a_j λ – 1 ), ( ν, λ = 0,1, … ; h = 1,2, … , m; j = 1,2, …, n), L is a suitable contour separating the simple poles of integrand h(s) in (2.2).

Now we define generalized fractional calculus operators as follows: Let and x > 0 then the fractional calculus operators in generalized form of arbitrary order involving Appell function F3, due to Saigo and Maeda [10] in the kernel are defined by the following equations:

(2.3)

(2.4)

(2.5)

(2.6)

The Saigo-Maeda operators are extensions of the Saigo operators defined as follows:

(2.7)

(2.8)

(2.9)

(2.10)

Following Chaurasia and Jain [18] we denote by u1 the class of functions f(x) on R₊ which are infinitely partialy differentiable with any order behaving as O(| x |^−ξ ) when x → ∞ for all ξ. Similarly by u₂, we denote the class of functions f(x,y) on the R₊ ^* R₊, which are infinitely partialy differentiable with any order behaving as O(| x |^−ξ1 | y |^−ξ2 ) when x → ∞, y → ∞ for all ξi ( i = 1,2 ).

On the same lines we denote by u₃, the class of functions f(x,y,z) defined on the R₊ * R₊ * R₊, which are infinitely partialy differentiable with any order behaving as O(| x |^−ξ1 | y |^−ξ2 | z |^−ξ3 ) when x → ∞, y → ∞, z → ∞ for all ξi ( i = 1,2,3 ).

The Saigo-Maeda operator of Weyl type three dimensional fractional integration of order Re(α_i) > 0, Re(γ_i) > 0 ( i = 1,2,3) is defined in the class u₃ by

(2.11)

THREE DIMENSIONAL LAPLACE TRANSFORM AND H-TRANSFORM: The Laplace transform h(p,q,r) of a function is written as [13] : let Re( p) > 0, Re(q) > 0, Re(r ) > 0

(3.1)

Similarly, the Laplace transform of

is defined by F(x,y,z), H(t) denotes Heaviside’s unit step function.

By the three dimensional H-transform ϕ ( p,q,r) of a function F(x,y,z), we mean the following repeated integral involving three different H-functions:

Here we assume that b > 0,d > 0, f > 0,k1 > 0,k2 > 0,k3 > 0; ϕ ( p,q,r) exists and belongs to u₃. The generality of the H-function, the equation (3.3) provides a generalization of a number of integral transforms like the three-dimensional Laplace, Stieltjes, Hankel, Whittaker and G-transforms.

RELATION BETWEEN THREE-DIMENSIONAL H-TRANSFORMS IN TERMS OF THREE-DIMENSIONAL SAIGO-MAEDA OPERATORS OF WEYL TYPE

In this section we evaluate three-dimensional H-transform ϕ1( p,q,r) of F(x,y,z) which will be needed for proof of the theorem considered in this section. We have

.F(x, y, z)dx dy dz (4.1)

Where it is assumed that ϕ₁( p,q,r) exists and belongs to u3, the parameters k₁ > 0, k₂ > 0, k₃ > 0 and other conditions on the additional parameters α₁,α'₁, β₁, β'₁,γ ₁,α₂,α'₂, β₂, β₂,γ'₂α₃,α'₃, β₃, β'₃,γ ₃, corresponding to those integral involved exists.

Theorem 4.1: Let ϕ ( p,q,r) be given by (3.3) then for Re(γ 1) > 0, Re(γ 2 ) > 0,Re(γ 3 ) > 0, d > 0, d > 0, f > 0, k1 > 0, k2 > 0, k3 > 0 there holds the formula

(4.2)

Where ϕ₁( p,q,r) is given by (4.1).

Proof: Let Re(γ 1) > 0, Re(γ 2 ) > 0,Re(γ 3 ) > 0, then in view of (3.3), we find that

ϕ (u,v,w)du dv dw

On interchanging the order of integration and evaluating the u, v, w-integrals, we get

Equation (4.4) can be established by means of the following formula [18]

(4.5)

represents the ratio of the product of several gamma functions i.e.

The left hand side of (4.3) becomes

Which is required right hand side of the equation (4.2).

As far as the three-dimensional Weyl type Saigo-Maeda operators preserve the class u₃, it follows that ϕ₁ ( p,q,r) also belongs to u₃.

SPECIAL CASES: By putting α'₁ =α'₂ =α'₃ = 0 , in theorem 4.1

and use the identity Theorem reduced in to the form of two dimensional and one dimensional analogue.

Acknowledgements

The authers are thanksful to the referee for the exhaustive comments for the improvement of the paper.

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