Integral Involving the Product of Multivariable Alephfunction, General Class of Srivastava Polynomials and Alephfunction of One Variable


Download PDF

Authors: D. KUMAR, F. AYANT AND NARESH

DOI: 10.46793/KgJMat2506.863K

Abstract:

In this paper, we derive an integral involving the multivariable Aleph-function, the general class of Srivastva polynomials, and the Aleph-function of one variable, all of which are sufficiently general in nature and are capable of yielding a large number of simpler and more useful results simply by specialization of their parameters. Moreover, we establish certain specific instances.



Keywords:

Aleph-function of several variables, general class of Srivastava polynomials, Aleph-function of one and two variables, I-function of two and several variables.



References:

[1]   F. Y. Ayant, An integral associated with the Aleph-functions of several variables, Int. J. Math. Trends Tech. 31(3) (2016), 142–154.

[2]   F. Y. Ayant and D. Kumar, A unified study of Fourier series involving the Aleph-function and the Kampé de Fériet’s function, Int. J. Math. Trends Tech. 35(1) (2016), 40–48. https://doi.org/10.14445/22315373/IJMTT/V35P507

[3]   F. Y. Ayant and D. Kumar, Certain finite double integrals involving the hypergeometric function and Aleph-function, Int. J. Math. Trends Tech. 35(1) (2016), 49–55. https://doi.org/10.14445/22315373/IJMTT/V35P508

[4]   F. Y. Ayant and D. Kumar, Generating relations and multivariable Aleph-function, Analysis 38(3) (2018), 137–143. https://doi.org/10.1515/anly-2017-0054

[5]   D. Baleanu, D. Kumar and S. D. Purohit, Generalized fractional integrals of product of two H-functions and a general class of polynomials, Int. J. Comput. Math. 93(8) (2016), 1320–1329. https://doi.org/10.1080/00207160.2015.1045886

[6]   V. B. L. Chaurasia and Y. Singh, New generalization of integral equations of fredholm type using Aleph-function, International Journal of Modern Mathematical Sciences 9(3) (2014), 208–220.

[7]   J. Choi and D. Kumar, Certain unified fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials, J. Inequal. Appl. 2014 (2014), 15 pages. https://doi.org/10.1186/1029-242X-2014-499

[8]   J. Daiya, J. Ram and D. Kumar, The multivariable H-function and the general class of Srivastava polynomials involving the generalized Mellin-Barnes contour integrals, FILOMAT 30(6) (2016), 1457–1464. https://doi.org/10.2298/FIL1606457D

[9]   I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1996.

[10]   R. K. Gupta, B. S. Shaktawat and D. Kumar, On generalized fractional differentials involving product of two H-functions and a general class of polynomials, J. Rajasthan Acad. Phys. Sci. 15(4) (2016), 327–344.

[11]   R. K. Gupta, B. S. Shaktawat and D. Kumar, Generalized fractional integrals and derivative formulas for H-function and a general class of polynomials, Journal of Chemical, Biological and Physical Sciences, Section C 6(3) (2016), 1081–1097.

[12]   D. Kumar, Fractional calculus formulas involving H-function and Srivastava polynomials, Commun. Korean Math. Soc. 31(4) (2016), 827–844. https://doi.org/10.4134/CKMS.c150251

[13]   D. Kumar, Generalized fractional differintegral operators of the Aleph-function of two variables, Journal of Chemical, Biological and Physical Sciences, Section C 6(3) (2016), 1116–1131.

[14]   D. Kumar, P. Agarwal and S. D. Purohit, Generalized fractional integration of the H-function involving general class of polynomials, Walailak Journal of Science and Technology 11(12) (2014), 1019–1030. https://doi.org/10.14456/WJST.2014.57

[15]   D. Kumar and F. Y. Ayant, Application of Jacobi polynomial and multivariable Aleph-function in heat conduction in non-homogeneous moving rectangular parallelepiped, Kragujevac J. Math. 45(3) (2021), 439–447. https://doi.org/10.46793/KgJMat2103.439K

[16]   D. Kumar and F. Y. Ayant, Fractional calculus operators pertaining to multivariable Aleph-function, Bulletin of Parana’s Mathematical Society 40 (2022), 1–10. https://doi.org/10.5269/bspm.42491

[17]   D. Kumar, F. Y. Ayant, S. Asawasamrit and J. Tariboon, Certain finite integrals related to the product of special functions, Symmetry 13 (2021), Article ID 2013, 11 pages. https://doi.org/10.3390/sym13112013

[18]   D. Kumar, F. Y. Ayant and J. Choi, Application of product of the multivariable A-function and the multivariable Srivastava polynomials, East Asian Math. J. 34(3) (2018), 295–303. https://doi.org/10.7858/eamj.2018.021

[19]   D. Kumar, F. Y. Ayant and D. Kumar, A new class of integrals involving generalized hypergeometric function and multivariable Aleph-function, Kragujevac J. Math. 44(4) (2020), 539–550. https://doi.org/10.46793/KgJMat2004.539K

[20]   D. Kumar, F. Y. Ayant and A. Prakash, Certain integral involving the product of Srivastava polynomials and special functions, Afr. Mat. 32 (2021), 1–9. https://doi.org/10.1007/s13370-021-00885-7

[21]   D. Kumar, S. D. Purohit and J. Choi, Generalized fractional integrals involving product of multivariable H-function and a general class of polynomials, J. Nonlinear Sci. Appl. 9 (2016), 8–21. https://doi.org/10.1007/s13370-021-00885-7

[22]   S. K. Kumari, V. T. M. Nambisan and A. K. Rathie, A study of I-functions of two variables, Le Matematiche 69(1) (2014), 285–305.

[23]   J. Ram and D. Kumar, Generalized fractional integration of the -function, J. Rajasthan Acad. Phys. Sci. 10(4) (2011), 373–382.

[24]   R. K. Saxena and D. Kumar, Generalized fractional calculus of the Aleph-function involving a general class of polynomials, Acta Math. Sci. Ser. B (Engl. Ed.) 35(5) (2015), 1095–1110. https://doi.org/10.1016/S0252-9602(15)30042-4

[25]   R. K. Saxena, J. Ram and D. Kumar, Generalized fractional integral of the product of two Aleph-functions, Appl. Appl. Math. 8(2) (2013), 631–646.

[26]   C. K. Sharma and S. S. Ahmad, On the multivariable I-function, Acta Ciencia Indica (Mathematics) 20(2) (1994), 113–116.

[27]   C. K. Sharma and P. L. Mishra, On the I-function of two variables and its properties, Acta Ciencia Indica (Mathematics) 17 (1991), 667–672.

[28]   K. Sharma, On the integral representation and applications of the generalized function of two variables, Int. J. Math. Eng. Sci. 3 (2014), 1–13.

[29]   H. M. Srivastava, A multi-linear generating function for the Konhauser set of biorthogonal polynomials suggested by Laguerre polynomial, Pacific. J. Math. 177 (1985), 183–191.

[30]   N. Südland, B. Baumann and T. F. Nonnenmacher, Open problem: who knows about the Aleph-functions?, Fract. Calc. Appl. Anal. 1(4) (1998), 401–402.