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SOME PROPERTIES OF EXTENDED τ-HYPERGEOMETRIC FUNCTION

  • Jana, Ranjan Kumar (Department of Applied Mathematics & Humanities S. V. National Institute of Technology) ;
  • Maheshwari, Bhumika (Department of Applied Mathematics & Humanities S. V. National Institute of Technology) ;
  • Shukla, Ajay Kumar (Department of Applied Mathematics & Humanities S. V. National Institute of Technology)
  • Received : 2017.08.04
  • Accepted : 2017.11.30
  • Published : 2018.10.31

Abstract

Recently, Parmar [5] introduced a new extension of the ${\tau}$-Gauss hypergeometric function $_2R^{\tau}_1(z)$. The main object of this paper is to study this extended ${\tau}$-Gauss hypergeometric function and obtain its properties including connection with modified Bessel function of third kind and extended generalized hypergeometric function, several contiguous relations, differential relations, integral transforms and elementary integrals. Various special cases of our results are also discussed.

Keywords

References

  1. G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999.
  2. M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994), no. 1, 99-124. https://doi.org/10.1016/0377-0427(94)90187-2
  3. M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman & Hall/CRC, Boca Raton, FL, 2002.
  4. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.
  5. R. K. Parmar, Extended $\tau$-hypergeometric functions and associated properties, C. R. Math. Acad. Sci. Paris 353 (2015), no. 5, 421-426. https://doi.org/10.1016/j.crma.2015.01.016
  6. E. D. Rainville, Special Functions, The Macmillan Co., New York, 1960.
  7. S. B. Rao, A. D. Patel, J. C. Prajapati, and A. K. Shukla, Some properties of generalized hypergeometric function, Commun. Korean Math. Soc. 28 (2013), no. 2, 303-317. https://doi.org/10.4134/CKMS.2013.28.2.303
  8. S. B. Rao and A. K. Shukla, Note on generalized hypergeometric function, Integral Transforms Spec. Funct. 24 (2013), no. 11, 896-904. https://doi.org/10.1080/10652469.2013.773327
  9. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
  10. H. M. Srivastava, A. Cetinkaya, and I. Onur Kymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 (2014), 484-491.
  11. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1984.
  12. N. Virchenko, S. L. Kalla, and A. Al-Zamel, Some results on a generalized hypergeometric function, Integral Transform. Spec. Funct. 12 (2001), no. 1, 89-100. https://doi.org/10.1080/10652460108819336
  13. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, reprint of the fourth (1927) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.