A New Class of Integrals Involving Generalized Hypergeometric Function and Multivariable Aleph-Function.


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Authors: D. KUMAR, F. AYANT AND D. KUMAR

DOI: 10.46793/KgJMat2004.539K

Abstract:

The aim of this paper is to evaluate an interesting integral involving generalized hypergeometric function and the multivariable Aleph-function. The integral is evaluated with the help of an integral involving generalized hypergeometric function obtained recently by Kim et al. [?]. The integral is further used to evaluate an interesting summation formula concerning the multivariable Aleph-function. A few interesting special cases and corollaries have also been discussed.

Keywords:

Multivariable I-function, multivariable H-function, double finite integrals.

References:

[1]   F. Ayant, An integral associated with the aleph-functions of several variables, International Journal of Mathematics Trends and Technology 31 (2016), 142–154.

[2]   F. Ayant and D. Kumar, Certain finite double integrals involving the hypergeometric function and aleph-function, International Journal of Mathematics Trends and Technology 35 (2016), 49–55.

[3]   J. Choi, J. Daiya, D. Kumar and R. Saxena, Fractional differentiation of the product of appell function F3 and multivariable H-function, Commun. Korean Math. Soc. 31 (2016), 115–129.

[4]   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), 1457–1464.

[5]   C. Fox, The G and H-functions as symmetrical fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395–429.

[6]    B. Gautam and A. Asga, The A-Function, Revista Mathematica, Tucuman, 1980.

[7]   B. Gautam and A. Asga, On the multivariable A-function, Vijnana Parishas Anusandhan Patrika 29 (1986), 67–81.

[8]   Y. Kim, M. Rakha and A. Rathie, Extension of certain classical summations theorems for the series 2F1,3F2 and 4F3 with applications in ramanujan’s summations, Int. J. Math. Sci. 2010 (2010), Article ID 309503, 26 pages.

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

[10]   D. Kumar, S. 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.

[11]   D. Kumar and J. Singh, Application of generalized M-series and H-function in electric circuit theory, MESA 7 (2016), 503–512.

[12]   K. Kumari, T. V. Nambisan and A. Rathie, A study of I-function of two variables, Le Matematiche 69 (2014), 285–305.

[13]   Y. Prasad, Multivariable I-function, Vijnana Parishad Anusandhan Patrika 29 (1986), 231–237.

[14]   Y. Prasad and A. Singh, Basic properties of the transform involving and H-function of r-variables as kernel, Indian Acad Math. 2 (1982), 109–115.

[15]   J. Prathima, V. Nambisan and S. Kurumujji, A study of I-function of several complex variables, International Journal of Engineering Mathematics 2014 (2014), 1–12.

[16]   J. Ram and D. Kumar, Generalized fractional integration involving appell hypergeometric of the product of two H-functions, Vijanana Parishad Anusandhan Patrika 54 (2011), 33–43.

[17]   V. Rohira, K. Kumari and A. Rathie, A new class of integral involving generalized hypergeometric function and the H-function, International Journal of Latest Engineering Research and Applications 2 (2017), 5–9.

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

[19]   V. Saxena, Formal solution of certain new pair of dual integral equations involving H-function, Proc. Nat. Acad. Sci. India Sect. A 51 (2001), 366–375.

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

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

[22]   J. Singh and D. Kumar, On the distribution of mixed sum of independent random variables one of them associated with srivastava’s polynomials and H-function, J. Appl. Math. Stat. Inform. 10 (2014), 53–62.

[23]   H. Srivastava, K. Gupta and S. Goyal, The H-Function of One and Two Variables with Applications, South Asian Publications, New Delhi, Madras, 1982.

[24]   H. Srivastava and R. Panda, Some expansion theorems and generating relations for the H-function of several complex variables, Comment. Math. Univ. St. Paul. 25 (1976), 119–137.

[25]   H. Srivastava and R. Panda, Some expansion theorems and generating relations for the H-function of several complex variables II, Comment. Math. Univ. St. Paul. 25 (1976), 167–197.

[26]   N. Südland, B. Baumann and T. Nonnenmacher, Open problem: who knows about the aleph-function? Fract. Calc. Appl. Anal. 1 (1998), 401–402.

[27]   N. Südland, B. Baumann and T. Nonnenmacher, Fractional drift-less fokker-planck equation with power law diffusion coefficients, in: V. G. Gangha, E. W. Mayr and E. V. Vorozhtsov (Eds.), Computer Algebra in Scientific Computing (CASC Konstanz 2001), Springer-Verlag, Berlin, 2001, 513–525.