Journal of Applied Mathematics
Volume 2019 (2019), Article ID 6821797, 10 pages
https://doi.org/10.1155/2019/6821797

Research Article

Finite Integral Formulas Involving Multivariable Aleph-Functions

Hagos Tadesse, D. L. Suthar, and Minilik Ayalew

Department of Mathematics, Wollo University, Dessie, P.O. Box 1145, Amhara Region, Ethiopia

Correspondence should be addressed to D. L. Suthar; dlsuthar@gmail.com

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 as

where ,

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:

where

with ; ; 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 as

And 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 by

where is the classical hypergeometric function. By substituting , the Jacobi polynomial (24) reduces to Lagrange polynomial ( [14], p. 157) as

In 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 becomes

Thus, , , , and

Now, using (21), we have the following.

Finally, we arrive at

Thus, , , and

Using (21), we have the following.

Finally, rewriting the above equation by virtue of (1), we arrive at

Thus, , , 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 obtain

Thus, , , and

Next we use the following formula ([16], Sec. 3.12) for

Using (42), we obtain

Thus, , , 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 obtain

Thus, , and .

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

The general class of polynomials defined by Srivastava [19] is as follows

Here ; ; 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

These 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 1–3 have been investigated in the literature by a number of authors [21–24]) 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.

References

C. K. Sharma and S. S. Ahmad, “On the multivariable I-function,” Acta Ciencia Indica. Mathematics, vol. 20, no. 2, pp. 113–116, 1994.

Y. N. Prasad, “Multivariable I-function,” Vijnana Parishad Anusandhan Patrika, vol. 29, pp. 231–235, 1986.

C. K. Sharma and G. K. Mishra, “Exponential Fourier series for the multivariable I-function,” Acta Ciencia Indica Mathematics, vol. 21, no. 4, pp. 495–501, 1995.

H. M. Srivastava and R. Panda, “Some expansion theorems and generating relations for the H function of several complex variables,” Commentarii Mathematici Universitatis Sancti Pauli, vol. 24, no. 2, pp. 119–137, 1975.

N. Sudland, B. Baumann, and T. F. Nonnenmacher, “Open problem: Who knows about the Aleph (?) -functions ?” Fractional Calculus and Applied Analysis, vol. 1, no. 4, pp. 401-402, 1998.

D. Kumar, S. D. Purohit, and J. Choi, “Generalized fractional integrals involving product of multivariable H-function and a general class of polynomials,” Journal of Nonlinear Sciences and Applications, vol. 09, no. 01, pp. 8–21, 2016.

R. K. Saxena and T. K. Pogány, “On fractional integration formulae for Aleph functions,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 985–990, 2011.

R. K. Saxena, J. Ram, and D. Kumar, “Generalized fractional integral of the product of two Aleph-functions,” Applications and Applied Mathematics. An International Journal (AAM), vol. 8, no. 2, pp. 631–646, 2013.

R. K. Saxena, J. Ram, and D. L. Suthar, “Unified fractional derivative formulas for the multivariable H-function,” Vijnana Parishad Anusandhan Patrika, vol. 49, no. 2, pp. 159–175, 2006.

D. L. Suthar and P. Agarwal, “Generalized mittag -leffler function and the multivariable h-function involving the generalized mellin-barnes contour integrals,” Communications in Numerical Analysis, vol. 2017, no. 1, pp. 25–33, 2017.

D. L. Suthar, B. Debalkie, and M. Andualem, “Modified Saigo fractional integral operators involving multivariable H-function and general class of multivariable polynomials,” Advances in Difference Equations, vol. 2019, no. 1, article no. 213, 2019.

D. L. Suthar, H. Habenom, and H. Tadesse, “Certain integrals involving aleph function and Wright's generalized hypergeometric function,” Acta Universitatis Apulensis, no. 52, pp. 1–10, 2017.

D. L. Suthar, G. Reddy, and B. Shimelis, “On Certain Unified Fractional Integrals Pertaining to Product of Srivastava's Polynomials and N-Function,” International Journal of Mathematics Trends and Technology, vol. 47, no. 1, pp. 66–73, 2017.

E. D. Rainville, Special Functions, The Macmillan Company, New York, NY, USA, 1960.

G. Szego, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI, USA, 1959.

A. Erde'lyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. I, McGraw-Hill, New York-Toronto-London, 1953.

V. S. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301 of Pitman Research Notes in Mathematics, United States with John Wiley & Sons, Longman Scientific & Technical, New York, NY, USA, 1994.

V. P. Saxena, “Formal solution of certain new pair of dual integral equations involving H-functions,” Proceedings of the National Academy of Sciences, vol. A52, no. 3, pp. 366–375, 1982.

H. M. Srivastava, “A multilinear generating function for the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials,” Pacific Journal of Mathematics, vol. 117, no. 1, pp. 183–191, 1985.

H. M. Srivastava and N. P. Singh, “The integration of certain products of the multivariableH-function with a general class of polynomials,” Rendiconti del Circolo Matematico di Palermo , vol. 32, no. 2, pp. 157–187, 1983.

F. Ayant, “The integration of certains products of special functions and multivariable Aleph-functions,” International Journal of Mathematics Trends and Technology, vol. 31, no. 2, pp. 101–112, 2016.

D. Kumar, R. K. Saxena, and J. Ram, “Finite integral formulas involving aleph function,” Boletim da Sociedade Paranaense de Matemática, vol. 36, no. 1, pp. 177–193, 2018.

D. L. Suthar, S. Agarwal, and D. Kumar, “Certain integrals involving the product of Gaussian hypergeometric function and Aleph function,” Honam Mathematical Journal, vol. 41, no. 1, pp. 1–17, 2019.

D. L. Suthar and B. Debalkie, “A new class of integral relation involving Aleph-functions,” Surveys in Mathematics and its Applications, vol. 12, pp. 193–201, 2017.