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Mathematica Eterna

Mathematica Eterna
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Research - (2021)Volume 11, Issue 3

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

Singh D1* and Jain, R2
 
*Correspondence: Singh D, Department of Applied Mathematics, School of Sciences, ITM University, India, Email:

Author info »

Abstract

In this paper we have made an attempt to establish a triple integral relation between weyl type three dimensional Saigo-Maeda operator of fractional integration and the three dimensional H-transform. Some special cases have also been established.

Keywords

Laplace transform, generalized fractional integral operator, H-transform, Appell function F3.MSC2010: 26A33, 33C65, 44A10.

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 equation -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 equation -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

equation

equation

Where 0 ≤ n ≤ p, 0 ≤ m ≤ q are non-negative integers; Aj and Bj are positive; aj ( j=1,…,p) and bj (j=1,…, q) are real or complex, such that Aj ( bh + ν ) ≠ Bh ( aj 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 equation 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:

equation (2.3)

equation (2.4)

equation (2.5)

equation (2.6)

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

equation (2.7)

equation (2.8)

equation (2.9)

equation (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(αi) > 0, Re(γi) > 0 ( i = 1,2,3) is defined in the class u3 by

equation

equation

equation (2.11)

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

equation (3.1)

Similarly, the Laplace transform of

equation

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:

equation

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

equation

equation

.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'112,α'2, β2, β2,γ'2α3,α'3, β3, β'3 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

equation (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

equation

equation

equation

ϕ (u,v,w)du dv dw

equation

equation

equation

equation

equation

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

equation

equation

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

equation (4.5)

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

equation

The left hand side of (4.3) becomes

equationequation

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

As far as the three-dimensional Weyl type Saigo-Maeda operators equation 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 equation 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.

References

  1. A. K. Arora, R. K. Raina and C. L. Koul, On the two-dimensional Weyl fractional calculus associated with the Laplace transforms, C.R. Acad. Bulg. Sci. 38, 1985, 179-182.
  2. A. M. Mathai, and R. K.Saxena, Generalized Hypergeometric Functions with Applications in Statistics and physical Sciences, Lecture Notes in Math., Springe, Berlin-Heidelberg-New York, 348, 1973.
  3. A. M. Mathai, Special Functions and Functions of Matrix Argument: Recent Advances and Applications in Stochastic Processes, Statistics Wavelet Analysis and Astrophysics, Lectuere Notes, CMS Pala Campus, Kerala, India, 34, 2007.
  4. C. Fox, The G and H-functions as symmetric Fourier kernels, Trans. Amer. Math. Soc., 98, 1961, 395-429.
  5. Dinesh Singh and Renu Jain, Fractional kinetic equation with a R. K. Raina, and V. Kiryakova, On the Weyl fractional operator of two-dimensions, C.R. Acad. Bulg. Sci. 36, 1983, 1273-1276.
  6. R. K. Saxena, and J. Ram, and D. L. Suthar, On two-dimensional Saigo- Maeda fractional calculus involving two-dimensional H-transforms, Acta. Ciencia Indica, XXXM, 4, 2004, 813-822.
  7. R. K. Saxena, and J. Ram, and J. D. Goree, On the two-dimensional Saigo fractional calculus associated with two-dimensional generalized H-transforms, Ganita Sandesh, 18 (1), 2004, 1-8.
  8. R. K. Saxena, and J. Ram, On the two-dimensional Whittaker transform, Serdica, Bulg. Math. Publ., 16, 1990, 27-30.
  9. R. K. Saxena, O. P. Gupta, and R. K. Kumbhat, On two-dimensional Weyl fractional calculus, Comptes rendus de Iâ??Academie des Sciences, Tome 42, N0 7, 1989, 11-14.
  10. Renu Jain, and Dinesh Singh, Certain fractional calculus operators associated with Fox-Wright generaliged hypergeometrin function, jnanbha, Vol.41, 2011, 1-8.
  11. V. B. L. Chaurasia, and m. Jain, Three-dimensional generalized Weyl fractional calculus pertaining to three-dimensional -transforms, Scientia, Series A: Mathematical Science, 19, 2010, 57-68.
  12. V. B. L. Chaurasia, and A. Shivastava, Two-dimensional generalized Weyl fractional calculus pertaining to two-dimensional -transforms, Tamkang J. of Math., 37, 2006, 237-249.

Author Info

Singh D1* and Jain, R2
 
1Department of Applied Mathematics, School of Sciences, ITM University, India
2Vice-Chancellor, Devi Ahilya Vishwavidyalaya Indore, India
 

Citation: Singh, D & Jain, R (2021). On Three-Dimensional Saigo-Maeda Operator of Weyl Type with Three-Dimensional H-Tran. Mathematica Eterna 11:129. doi: 10.35248/1314-3344.21.11.129

Received: 25-Apr-2021 Accepted: 05-Jun-2021 Published: 09-Jun-2021 , DOI: 10.35248/1314-3344.21.11.129

Copyright: © 2021. Singh, D & Jain, R This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited.

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