Journal of Applied Mathematics
Volume 2019, Article ID 6821797, 10 pages
https://doi.org/10.1155/2019/6821797
Finite Integral Formulas Involving Multivariable Aleph-Functions
Department of Mathematics, Wollo University, Dessie, P.O. Box 1145, Amhara Region, Ethiopia
Correspondence should be addressed to D. L. Suthar; moc.liamg@rahtusld
Received 6 March 2019; Revised 10 July 2019; Accepted 10 July 2019; Published 21 August 2019
Academic Editor: Kai Diethelm
Copyright © 2019 Hagos Tadesse et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 [1–3]]).
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 [6–13].
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 have