Journal of Applied Mathematics

Journal of Applied Mathematics / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 6821797 | 10 pages |

Finite Integral Formulas Involving Multivariable Aleph-Functions

Academic Editor: Kai Diethelm
Received06 Mar 2019
Revised10 Jul 2019
Accepted10 Jul 2019
Published21 Aug 2019


The integrals evaluated are the products of multivariable Aleph-functions with algebraic functions, Jacobi polynomials, Legendre functions, Bessel-Maitland functions, and general class of polynomials. The main results of our paper are quite general in nature and competent at yielding a very large number of integrals involving polynomials and various special functions occurring in the problem of mathematical analysis and mathematical physics.

1. Introduction and Preliminaries

Throughout this paper, consider , and to be a set of complex numbers, positive real numbers, nonpositive integers, and positive integers, respectively. The multivariable Aleph function of several complex variables generalizes the multivariable I-function, recently studied by Sharma and Ahmad [1], which itself is a generalization of G- and H-functions of multiple variables aswhere ,and ; ; and are complex numbers. are positive real numbers.

The integration path extends from to and the poles of , do not coincide with the poles of and to the left of the contour .

The existence condition for multiple Mellin-Barnes type contours (1) can be given below: wherewith ; ; and

Remark 1. By setting , the multivariable Aleph-function reduces to multivariable I-function (see [13]]).

Remark 2. By setting and , the multivariable Aleph-function reduces to multivariable H-function defined by Srivastava et al. [4].

Remark 3. When we set , the multivariable Aleph-function reduces to Aleph-function of one variable defined by Sdland [5].

For the definition of the H- function, -function, and its more generalization, the interested reader may refer to the papers [613].

From Rainville [14], the integral representation of the gamma function is defined asAnd also the beta integral is defined as follows:Further, We will use the following notations in this paper:

2. Integrals Involving Multivariable Aleph-Function with Algebraic Function

In this section, we evaluate integrals, the product of multivariable Aleph-functions with various algebraic functions.Putting , we have , and we use the following relation.Setting implies , and then we obtain the following. By applying (14), we have the following.Now, we use the following formula ([14], p.261).Hence, we arrive at

3. Integrals Involving Multivariable Aleph-Function with Jacobi Polynomials

The Jacobi polynomial ([15], 4.21.2)) with parameters is defined bywhere is the classical hypergeometric function. By substituting , the Jacobi polynomial (24) reduces to Lagrange polynomial ( [14], p. 157) asIn this section, we derive integral formulas involving multivariable Aleph-functions multiplied by Jacobi polynomials.Next, we use the following formula:where and Also, is the special case of generalized hypergeometric series.

Then, we have the following.Using the definition of hypergeometric function and some simplifications, the above expression becomesThus, , , , and Now, using (21), we have the following.Finally, we arrive atThus, , , and Using (21), we have the following.Finally, rewriting the above equation by virtue of (1), we arrive atThus, , , and

4. Integrals Involving Multivariable Aleph-Function with Legendre Function

The solution of Legendre differential equation in the form of Gauss hypergeometric type is as follows:Here is known as the Legendre function of the first kind [16].

Next, we derive the integrals with Legendre function.Next, we use the following formula ([16], Sec. 3.12) for Now, applying (38), we obtainThus, , , and Next we use the following formula ([16], Sec. 3.12) for Using (42), we obtainThus, , , and

5. Integrals Involving Multivariable Aleph-Function and Bessel-Maitland Function

The Bessel-Maitland function is defined by Kiryakova [17] as follows: