On Transformation Involving Basic Analogue of Multivariable -Function

Abstract

In this article, fractional order -integrals and -derivatives involving a basic analogue of multivariable -function have been obtained. We give an application concerning the basic analogue of multivariable -function and -extension of the Leibniz rule for the fractional -derivative for a product of two basic functions. We also give the corollary concerning basic analogue of multivariable Meijer’s -function as a particular case of the main result.

1. Introduction and Preliminaries

The -calculus is not of recent appearance, it was introduced in the twenties of last century. In 1910, Jackson [1] introduced and developed -calculus systematically. The fractional -calculus is the expansion of ordinary fractional calculus in the -theory. Recently, there was a significant work done by many authors in the area of -calculus due to lots of applications in mathematics, statistics, and physics.

Since special functions play significant roles in mathematical physics, it is persuaded to think that some deformation of the ordinary special functions based on the -calculus can also play comparable roles in this area of research. Further, many authors have derived images of various -special functions under fractional -calculus operators; see, for example, [2–7], and may more. The -fractional integrals and derivatives was firstly studied by Al-Salam [8] (see also, [9]). Many researchers have used these operators to evaluate fractional -calculus formulas for various special function, general class of -polynomials, basic analogue of Fox’s -function, fractional -calculus formulas for various special function, and etc. One may refer to the recent work [2–7, 10–14] on fractional -calculus. Throughout this article, let , and be the sets of integers, complex numbers, real numbers, positive real numbers, and positive integers, respectively, and let. .

The objective of this article is to establish fractional -integral and -derivative of Riemann-Liouville type involving a basic analogue of multivariable -function. We also give an application of -Leibniz formula.

In the -calculus theory, for a real parameter , we have a -real number

and -shifted factorial (-analogue of the Pochhammer symbol) as given by

The -Factorial function is defined by

Its extension is which can be elaborated to , given by where the principal value of is taken.

In terms of the -gamma function, (2) can be written as where the -gamma function [15] is given by obviously,

The -analogue of the familiar Riemann-Liouville fractional integral operator of a function is defined by (see Al-Salam [8]) also -analogue of the power function is defined as

The basic integral is given by (see Gasper and Rahman [15])

The equation (8) in conjunction with (10) yield the following series representation of the Riemann-Liouville fractional integral operator

In particular, for , we have [4]

2. Basic Analogue of Multivariable -Function

In this section, we introduce the basic analogue of multivariable -function [16, 17], given by the following manner: where , and here, and

The integers are constrained by the inequalities and . The poles of integrand are assumed to be simple. The quantities are complex numbers and the following quantities are positive real numbers.

The contour in the complex -plane is of the Mellin-Barnes type which runs from to with indentations, if necessary to make certain that all the poles of are separated from those of, . For large values of , the integrals converge if .

If the quantities , then the basic analogue of multivariable -function reduces in basic analogue of multivariable Meijer’s -function defined by Khadia and Goyal [18], we obtain where where ; the integers are constrained by the inequalities and . The poles of integrand are assumed to be simple. The quantities are complex numbers.

The contour in the complex -plane is of the Mellin-Barnes type which runs from to with indentations, if necessary to ensure that all the poles of are separated from those of . For large values of , the integrals converge if .

3. Main Results

In this section, we establish two fractional -integral formulas about the basic analogue of multivariable -function.

Let

Theorem 1. Let, the Riemann-Liouville fractional-integral of a product of two basic functions exists, and we havewhere .

Proof. To prove the result (19), we consider the left hand side of equation (19) (say I) and take the definitions (8) and (13) into account, we have Interchanging the order of integrations which is permissible under the given conditions, we obtain The above equation writes Now using the result (12), then the equation (22) reduces as Next, interpreting the -Mellin-Barnes multiple integrals contour in terms of the basic analogue of multivariable -function, then we get the desired result (19).
If we replace by in Theorem 1, and use the fractional - derivative operator defined as and power function formula then we have the following result:

Theorem 2. Letthe Riemann Liouville fractional-derivative of a product of two basic functions exists, and given bywhere.

Proof. The proof of result asserted by Theorem 2 runs parallel to that of Theorem 1.
The details are, therefore, being omitted.

4. Leibniz’s Application

In this section, we give an application concerning the basic analogue of multivariable -function and -extension of the Leibniz rule for the fractional -derivative for a product of two basic functions.

We have the -extension of the Leibniz rule for the fractional -derivatives for a product of two basic functions in terms of a series involving the fractional -derivatives of the function, in the following manner [9]:

Lemma 3. whereandare two regular functions.

Theorem 4. Let, then the Riemann-Liouville fractional-derivative of a product of two basic function exists and given bywhere .

Proof. For applying -Leibniz rule, we let By using the lemma 3, we have Next, by setting and using the Theorem 2, we arrive at where .

Now, by using (25) and (31) we obtain the desired result (28) after several algebraic manipulations.

5. Particular Case

In this section, the basic analogue of multivariable -function reduces in basic analogue of multivariable Meijer’s -function [18].

Let

Corollary 5. where .

Remark 6. If the basic analogue of multivariable-function reduces in basic analogue of Srivastava-Daout function [19], then we obtain the results given by Purohit et al. [20].

Remark 7. If the basic analogue of multivariable-function reduces in basic analogue of-function of two variables defined by Saxena et al. [21], we obtain the result due to Yadav et al. [7]. Further, if the basic analogue of multivariable-function reduces in basic analogue of-function of one variable defined by Saxena et al. [22], then we can easily obtain the similar result.

6. Conclusion

In the present article, we have proposed the fractional order -integrals and -derivatives involving a basic analogue of multivariable -function. The significance of our derived results lies in their diverse generality. By specializing the various parameters as well as variables in the basic analogue of multivariable -function, we can obtain a large number of results involving a remarkably wide range of useful basic functions (or product of such basic functions) of one and several variables. Hence, the derived formulas in this article are most general in character and may reaffirm to be useful in several interesting cases appearing in literature.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-61-KNOW-021.