Abstract

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:Now, using the following formula from [18]and, further, applying (47), we obtainThus, , and .

6. Integrals Involving Multivariable Aleph-Function and General Class of Polynomials

The general class of polynomials defined by Srivastava [19] is as followsHere ; ; and the coefficients are any constant numbers. By suitable restriction of the coefficient the general class of polynomials has various special cases. These include the Jacobi polynomials, the Laguerre polynomials, the Bessel polynomials, the Hermite polynomials, the Gould-Hopper polynomials, and the Brafman polynomials ([20], p. 158-161).

Next, we establish the following integral.By applying (21), we haveThese converge under the following conditions:(1),(2) ( and are not both zero simultaneously),(3) and

7. Special Cases

(i) By setting by and , (32) transforms to the following integral:Thus,

(ii) When we substitute by and by and put , integral (35) transforms to the following:Thus,

8. Concluding Remarks

In the present paper, we evaluated new integrals involving the multivariable Aleph-function with certain special functions. Certain special cases of integrals that follow Remarks 13 have been investigated in the literature by a number of authors [2124]) with different arguments. Therefore, the results presented in this paper are easily converted in terms of a similar type of new interesting integrals with different arguments after some suitable parameter substitutions. In this sequel, one can obtain integral representation of more generalized special function, which has many applications in physics and engineering science.

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.