• Choi, Junesang (Department of Mathematics, Dongguk University) ;
  • Rathie, Arjun K. (Department of Mathematics, School of Mathematical & Physical Sciences, Central University of Kerala, Riverside Transit Campus) ;
  • Parmar, Rakesh K. (Department of Mathematics, Government College of Engineering and Technology)
  • Received : 2014.03.13
  • Accepted : 2014.04.23
  • Published : 2014.06.25


Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the confluent hypergeometric function by using their integral representations and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the abovementioned extended functions and investigating various formulas for the further extended functions in a systematic manner. Moreover, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.


Gamma function;Beta function;hypergeometric function;extended Beta function;extended hypergeometric function;extended confluent hypergeometric function;Mellin transform;summation formula


  1. M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, CRC Press (Chapman and Hall), Boca Raton, FL, 2002.
  2. M. A. Chaudhry and S. M. Zubair, Generalized incomplete gamma functions with applications, J. Comput. Appl. Math. 55 (1994), 99-124.
  3. M. A. Chaudhry and S. M. Zubair, On the decomposition of generalized incom-plete gamma functions with applications to Fourier transforms, J. Comput. Appl. Math. 59 (1995), 253-284.
  4. M. A. Chaudhry and S. M. Zubair, Extended incomplete gamma functions with applications, J. Math. Anal. Appl. 274 (2002), 725-745.
  5. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Inte-gral Transforms, Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1954.
  6. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Corrected and Enlarged edition prepared by A. Je rey), sixth edition, Academic Press, New York, 2000.
  7. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  8. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, London, and New York, 1966.
  9. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
  10. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London, and New York, 2012.
  11. H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, Madras, 1982.
  12. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
  13. M. A. Chaudhry, N. M. Temme, and E. J. M. Veling, Asymptotic and closed form of a generalized incomplete gamma function, J. Comput. Appl. Math. 67 (1996), 371-379.
  14. Y. A. Brychkov, Handbook of Special Functions (Derivatives, Integrals, Series and Other Formulas), Taylor & Francis Group, LLC; Chapman & Hall/CRC, 2008.
  15. M. A. Chaudhry, A. Qadir, M. Ra que, and S. M. Zubair, Extension of Euler's Beta function, J. Comput. Appl. Math. 78 (1997), 19-32.
  16. M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, Extended hy-pergeometric and con uent hypergeometric functions, Appl. Math. Comput. 159 (2004), 589-602.

Cited by

  1. On extended Hurwitz–Lerch zeta function vol.448, pp.2, 2017,
  2. On properties and applications of ( p , q )-extended τ -hypergeometric functions 2018,
  3. A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators vol.2018, pp.1, 2018,