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CERTAIN INTEGRALS INVOLVING THE PRODUCT OF GAUSSIAN HYPERGEOMETRIC FUNCTION AND ALEPH FUNCTION

Suthar, D.L.;Agarwal, S.;Kumar, Dinesh

  • Received : 2017.05.28
  • Accepted : 2019.01.21
  • Published : 2019.03.25

Abstract

The aim of this paper is to establish certain integrals involving product of the Aleph function with exponential function and multi Gauss's hypergeometric function. Being unified and general in nature, these integrals yield a number of known and new results as special cases. For the sake of illustration, twelve corollaries are also recorded here as special case of our main results.

Keywords

Aleph function;Hypergeometric function;H-Function;I-function;Mellin-Barnes type contour integral

References

  1. L.K. Arora and U.K. Saha, Integrals involving Hypergeometric function and H-function, J. Indian Acad. Math., 32(1) (2010), 243-249.
  2. A. Bhargava, A. Srivastava and R. Mukherjee, Some integrals involving I-function and Wright's generalized Hypergeometric function, Casp. J. of App. Math., Ecology and Economics, 3(1) (2015), 3-11.
  3. 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., 2014:499, (2014), 1-15.
  4. 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.
  5. D. Kumar, Certain integrals of generalized hypergeometric and con uent hyper-geometric functions, Sigmae, 5(2) (2017), 8-18.
  6. D. Kumar and F.Y. Ayant, A unified study of Fourier series involving the Aleph-function and the Kampe de Feriet's function, Int. J. Math. Trends Technol., 35(1) (2016), 40-48. https://doi.org/10.14445/22315373/IJMTT-V35P507
  7. D. Kumar and J. Choi, Generalized fractional Kinetic equations associated with Aleph function, Proc. Jangjeon Math. Soc., 19(1) (2016), 145-155.
  8. D. Kumar, R.K. Gupta and B.S. Shaktawat, Certain results on extended generalized $\tau$-Gauss hypergeometric function, Honam Mathematical J., 38(4) (2016), 739-752. https://doi.org/10.5831/HMJ.2016.38.4.739
  9. D. Kumar, R.K. Gupta, B.S. Shaktawat and J. Choi, Generalized fractional calculus formulas involving the product of Aleph-function and Srivastava polynomials, Proc. Jangjeon Math. Soc., 20(4) (2017), 701-717.
  10. D. Kumar, R.K. Saxena and J. Ram, Finite integral formulas involving Aleph function, Bol. Soc. Parana. Mat., 36(1) (2018), 177-193. https://doi.org/10.5269/bspm.v36i1.28123
  11. A.M. Mathai, R.K. Saxena, and H.J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2010.
  12. E.D. Rainville, Special Functions, Chelsea Publishing Co., Bronx, New York, 1971.
  13. J. Ram and D. Kumar, Generalized fractional integration of the N-function, J. Raj. Acad. Phy. Sci., 10(4) (2011), 373-382.
  14. U.K. Saha, L.K. Arora and B.K. Dutta, Integrals involving I-function, Gen. Math. Notes, 6(1) (2011), 1-14.
  15. V.P. Saxena, Formal solution of certain new pair of dual integral equations involving H-functions, Proc. Nat. Acad. Sci. India Sect., A 51 (1982), 366-375.
  16. R.K. Saxena and D. Kumar, Generalized fractional calculus of the Aleph function involving a general class of polynomials, Acta Math. Sci. Ser. B, 35(5) (2015), 1095-1110. https://doi.org/10.1016/S0252-9602(15)30042-4
  17. R.K. Saxena and T.K. Pogany, Mathieu-type series for the Aleph-function occurring in Fokker-Planck equation, Eur. J. Pure Appl. Math., 3(6) (2010), 980-988.
  18. R.K. Saxena and T.K. Poany, On fractional integration formulae for Aleph functions, Appl. Math. Comput., 218 (2011), 985-990. https://doi.org/10.1016/j.amc.2011.03.026
  19. L.J. Slater, Generalized Hypergeometric functions, Cambridge University Press, 1966.
  20. N. Sudland, B. Baumann and T.F. Nannenmacher, Open problem: Who knows about the Aleph(N)-function, Appl. Anal., 1(4) (1998), 401-402.
  21. N. Sudland, B. Baumann and T.F. Nannenmacher, Fractional driftless Fokker-Planck equation with power law di usion coecients, in V.G. Gangha, E.W. Mayr, W.G. Vorozhtsov (Eds.), Computer Algebra in Scientific Computing (CASC Konstanz 2001) Springer, Berlin, (2001), 513-525.
  22. D.L. Suthar and S. Agarwal, Marichev-Saigo-Maeda fractional integration operator associated with Srivastava's polynomial and Gauss hypergeometric functions, Konuralp J. Math., 5(1) (2016), 145-160.