Abstract
Srivastava and Panda have studied the simple and multiple generating relations concerning the multivariable H-function. The aim of this paper is to derive the various classes of simple and multiple generating relations involving the multivariable Aleph-function. The generating function is used in the theory of numbers, in physics and other fields of mathematics. We see the particular cases concerning the multivariable I-function, the Aleph-function of two variables and the I-function of two variables.
Funding statement: Dinesh Kumar would like to express his deep thanks to NBHM (National Board of Higher Mathematics), India, for granting a Post-Doctoral Fellowship (sanction no. 2/40(37)/2014/R&D-II/14131).
References
[1] P. Agarwal, M. Chand, I. O. Kıymaz and A. Çetinkaya, A certain sequence of functions involving the Aleph function, Open Phys. 14 (2016), 187–191. 10.1515/phys-2016-0018Search in Google Scholar
[2] P. Agarwal, J. Choi and S. Jain, Extended hypergeometric functions of two and three variables, Commun. Korean Math. Soc. 30 (2015), no. 4, 403–414. 10.4134/CKMS.2015.30.4.403Search in Google Scholar
[3] P. Agarwal, F. Qi, M. Chand and S. Jain, Certain integrals involving the generalized hypergeometric function and the Laguerre polynomials, J. Comput. Appl. Math. 313 (2017), 307–317. 10.1016/j.cam.2016.09.034Search in Google Scholar
[4] F. Ayant, An integral associated with the Aleph-functions of several variables, Int. J. Math. Trends Technol. 31 (2016), no. 3, 142–154. 10.14445/22315373/IJMTT-V31P522Search in Google Scholar
[5] 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 Technol. 35 (2016), no. 1, 40–48. 10.14445/22315373/IJMTT-V35P507Search in Google Scholar
[6] F. Y. Ayant and D. Kumar, Certain finite double integrals involving the hypergeometric function and Aleph-function, Int. J. Math. Trends Technol. 35 (2016), no. 1, 49–55. 10.14445/22315373/IJMTT-V35P508Search in Google Scholar
[7] J. Choi and P. Agarwal, Certain class of generating functions for the incomplete hypergeometric functions, Abstr. Appl. Anal. 2014 (2014), Article ID 714560. 10.1155/2014/714560Search in Google Scholar
[8]
J. Choi, J. Daiya, D. Kumar and R. K. Saxena,
Fractional differentiation of the product of Appell function
[9] 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 (2016), no. 6, 1457–1464. 10.2298/FIL1606457DSearch in Google Scholar
[10] R. K. Gupta, B. S. Shaktawat and D. Kumar, On generalized fractional derivative involving product of two H-functions and a general class of polynomials, J. Rajasthan Acad. Phys. Sci. 15 (2016), no. 4, 327–344. Search in Google Scholar
[11] D. Kumar, Certain integrals of generalized hypergeometric and confluent hypergeometric functions, Sigmae 5 (2016), no. 2, 8–18. Search in Google Scholar
[12] D. Kumar, Generalized fractional differintegral operators of the Aleph-function of two variables, J. Chem. Biol. Phys. Sci. Sect. C 6 (2016), no. 3, 1116–1131. Search in Google Scholar
[13]
D. Kumar and J. Choi,
Certain generalized fractional differentiation of the product of two
[14] D. Kumar and J. Choi, Generalized fractional kinetic equations associated with Aleph functions, Proc. Jangjeon Math. Soc. 19 (2016), no. 1, 145–155. Search in Google Scholar
[15] 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), no. 1, 8–21. 10.22436/jnsa.009.01.02Search in Google Scholar
[16] D. Kumar, R. K. Saxena and J. Ram, Finite integral formulas involving aleph function, Bol. Soc. Parana. Mat. (3) 36 (2018), no. 1, 177–193. 10.5269/bspm.v36i1.28123Search in Google Scholar
[17] G. Pólya and G. Szegő, Problems and Theorems in Analysis. I, Classics Math., Springer, Berlin, 1998. 10.1007/978-3-642-61905-2Search in Google Scholar
[18]
J. Ram and D. Kumar,
Generalized fractional integration of the
[19] 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 (2015), no. 5, 1095–1110. 10.1016/S0252-9602(15)30042-4Search in Google Scholar
[20] R. K. Saxena, J. Ram and D. Kumar, Generalized fractional integral of the product of two Aleph-functions, Appl. Appl. Math. 8 (2013), no. 2, 631–646. Search in Google Scholar
[21] C. K. Sharma and S. S. Ahmad, On the multivariable I-function, Acta Cienc. Indica Math. 20 (1994), no. 2, 113–116. Search in Google Scholar
[22] C. K. Sharma and P. L. Mishra, On the I-function of two variables and its certain properties, Acta Cienc. Indica Math. 17 (1991), no. 4, 667–672. Search in Google Scholar
[23] K. Sharma, On the integral representation and applications of the generalized function of two variables, Int. J. Math. Eng. Sci. 3 (2014), 1–13. Search in Google Scholar
[24] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, John Wiley & Sons, New York, 1984. Search in Google Scholar
[25] H. M. Srivastava and R. Panda, Some expansion theorems and generating relations for the H function of several complex variables, Comment. Math. Univ. St. Pauli 24 (1975/76), no. 2, 119–137. Search in Google Scholar
[26] H. M. Srivastava and R. Panda, Some expansion theorems and generating relations for the H function of several complex variables. II, Comment. Math. Univ. St. Pauli 25 (1976/77), no. 2, 167–197. Search in Google Scholar
[27]
N. Südland, B. Baulmann and T. F. Nonnenmacher,
Open problem: Who knows about the Aleph (
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